# Quantum Entropic Relationalism (QER) Contemporary debates and theoretical frontiers

Abdelouahab Rgoud

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Article

# Quantum Entropic Relationalism (QER)

## Contemporary debates and theoretical frontiers

Abdelouahab RGOUD<fnref n="1" />

**Abstract**

Entropy has evolved from a secondary thermodynamic property (Clausius, 1865) to a potentially fundamental organizing structure of physical reality, particularly through its gravitational manifestation in the Bekenstein-Hawking formula. This article systematically reviews four theoretical developments (2015-2024) that test this hypothesis using analytical methods from quantum information theory, holographic duality, and quantum gravity. First, we examine how the quantum island formula resolves the black hole information paradox by demonstrating that fine-grained entropy depends on global causal structure rather than local degrees of freedom. Second, we analyze the Complexity=Volume and Complexity=Action conjectures, showing that computational complexity encodes post-thermalization dynamics on exponentially long timescales, with predicted maximum complexity $C_{max} \sim S_{BH}^{3/2}$ testable in SYK simulations. Third, we compare three unification programs (AdS/CFT holography, emergent gravity, loop quantum gravity) and construct their distinct observational predictions. Fourth, we explore entropy-motion duality through mixed metric signatures, arguing that the correspondence $\beta \leftrightarrow it$ reveals deep physical structure. We articulate quantum entropic relationalism as an epistemological framework wherein entropy constitutes an objective relational structural property encoding physical relations without substantial reducibility. This synthesis suggests spacetime emerges from quantum entanglement substrates, with testability prospects via gravitational interferometry (LISA, Einstein Telescope), quantum simulators, and cosmological observations (CMB-S4, LiteBIRD) anticipated by 2035.

**Keywords:** entropic relationalism, quantum entanglement, holography, emergent geometry, computational complexity, informational gravitation

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**Citation:** . 2025. . *Journal* 1: 0.

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**Copyright:** © 2025 by the authors.

Submitted to *Journal* for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

<footnote n="1">¹ ACPPS-UM6P, abdelouahab.rgoud@um6p.ma</footnote>

<footnote>* Correspondence: abdelouahab.rgoud@um6p.ma</footnote>
# Introduction

Entropy, from its thermodynamic formulation by Clausius in 1865 to its contemporary incarnations in quantum gravity, is one of the most powerful and enigmatic concepts in theoretical physics. Its intellectual trajectory spans three major conceptual revolutions that have progressively transformed our understanding of the very nature of physical reality. The first revolution came with Boltzmann in 1877, who reinterpreted thermodynamic entropy according to the formula $S = k_B \ln \Omega$ as a measure of the number of microstates $\Omega$ compatible with a given macrostate<fnref n="1" />, establishing the fundamental link between statistical mechanics and thermodynamics. The second revolution emerged with Shannon in 1948 and Jaynes in 1957, who demonstrated the formal equivalence between thermodynamic entropy and informational entropy $S = - \sum_i p_i \ln p_i^2$, revealing that entropy fundamentally quantifies our ignorance about the microscopic state of a system. The third revolution, inaugurated by Bekenstein in 1973 and Hawking in 1975, established that a black hole has entropy proportional to the area of its horizon, $S_{BH} = A/(4\ell_P^2)$, where $\ell_P = \sqrt{G\hbar/c^3}$ represents the Planck length<fnref n="3" />. This discovery unified thermodynamics, quantum mechanics, and gravity for the first time, suggesting that entropy potentially constitutes a primary organizing structure transcending traditional divisions between physical domains. This latest discovery catalyzed a radical epistemological transformation whose implications continue to unfold half a century later. Entropy ceases to be a secondary thermodynamic property and potentially becomes the very foundation from which space-time, energy-matter, and their dynamic laws emerge. This hierarchical reversal, comparable to the transition from ether as a primary substance to electromagnetic fields as fundamental entities, nevertheless raises three persistent puzzles that motivate a thorough epistemological reassessment.

## *Three fundamental puzzles*

The first puzzle concerns the ontological status of entropy, which is ambiguous to say the least. Boltzmann's formula depends essentially on *coarse-graining*, i.e., the partitioning of phase space into distinct macrostates. Two observers using different partitionings will necessarily assign different entropies to the same macroscopic state $|\psi\rangle$. Wheeler articulated this tension by asking whether entropy was not fundamentally anthropomorphic<fnref n="4" />. Jaynes attempted to resolve this tension by adopting a radical epistemological position, asserting that entropy measures exclusively our subjective ignorance<fnref n="5" />. However, this subjectivist interpretation faces insurmountable difficulties in the face of black hole entropy, whose value $S_{BH} = A/(4G\hbar)$ seems manifestly independent of any particular observer and encodes objective geometric properties of the horizon.

The second enigma emerges on a cosmological scale and concerns the origin of fundamental temporal asymmetry. The current observable universe has a total entropy $S_{\text{current}} \sim 10^{103} k_B$, dominated by supermassive black holes, but exhibited an extraordinarily low entropy $S_{\text{initial}} \sim 10^{88} k_B$ at the time of recombination<fnref n="6" />. Penrose calculates that the probability of spontaneously obtaining such an ordered configuration is of the order of

<footnote n="1">¹ L. Boltzmann, "Über die Beziehung zwischen dem zweiten Hauptsatze des mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung," *Wiener Berichte* **76**, 373 (1877).</footnote>

<footnote n="2">² C.E. Shannon, "A mathematical theory of communication," *Bell Syst. Tech. J.* **27**, 379-423 (1948); E.T. Jaynes, "Information theory and statistical mechanics," *Phys. Rev.* **106**, 620-630 (1957).</footnote>

<footnote n="3">³ J.D. Bekenstein, "Black holes and entropy," *Phys. Rev. D* **7**, 2333-2346 (1973); S.W. Hawking, "Particle creation by black holes," *Commun. Math. Phys.* **43**, 199-220 (1975).</footnote>

<footnote n="4">⁴ J.A. Wheeler, "Information, physics, quantum: The search for links," in *Complexity, Entropy, and the Physics of Information*, (1990).</footnote>

<footnote n="5">⁵ E.T. Jaynes, "The Gibbs paradox," in *Maximum Entropy and Bayesian Methods*, (1992).</footnote>

<footnote n="6">⁶ C.A. Egan and C.H. Lineweaver, "A larger estimate of the entropy of the universe," *Astrophys. J.* **710**, 1825 (2010).</footnote>
$10^{-10^{1237}}$, suggesting that a fundamental physical explanation is required. Three contemporary programs attempt to solve this puzzle by different means: the eternal inflationary multiverse<fnref n="8" />, Penrose’s conformal cyclic cosmology<fnref n="9" />, and the quantum bounce of loop quantum gravity<fnref n="10" />. These incompatible approaches reveal the lack of theoretical consensus regarding the cosmological origin of entropic asymmetry.

The third puzzle concerns the black hole information paradox and reveals an apparent incompatibility between quantum unitarity and the thermodynamics of horizons. When a black hole evaporates via Hawking radiation, the information encoded in the pure initial state appears to be irretrievably lost, violating the fundamental unitarity of quantum mechanics<fnref n="11" />. Page demonstrated in 1993 that if the evolution is unitary, the entropy of the external radiation must necessarily decrease after the “Page time” $t_{\text{Page}}^{12}$, without any physical mechanism allowing this decrease being known until 2019. The partial resolution via the quantum island formula, developed by Penington and by Almheiri, Engelhardt, Marolf, and Maxfield, unifies semi-classical geometry and quantum entanglement<fnref n="13" />. The entropy of a radiation system is calculated by including a geometric “island” $\mathcal{I}$ located inside the black hole, whose contribution depends on the overall causal structure. This discovery reveals that gravitational entropy is no longer a local property but emerges from a global geometric optimization that is intrinsically dependent on the causal structure of spacetime.

### Central thesis: Quantum Entropic Relationalism (QER)

These three enigmas converge toward a radical conceptual hypothesis that this article systematically develops and defends. Our central thesis asserts that gravitational entropy constitutes a relational quantity emerging from spatial partitions, causal structures, and quantum entanglement, thus transcending the traditional dichotomy between absolute objectivity and anthropomorphic subjectivity. This conceptual relativization, analogous to the transition from Newtonian absolute spacetime to Einsteinian general relativity, where simultaneity becomes relative to the reference frame without becoming subjective, constitutes the central epistemological contribution of this article.

Three arguments structure this relational thesis. The first concerns the spatial relationality revealed by the AdS/CFT correspondence. The entanglement entropy of a subregion $A$ is calculated via the area of an extremal surface $\gamma_A$ according to the Ryu-Takayanagi formula $S_A = \text{Area}(\gamma_A)/(4G\hbar)^{14}$. In particular, $\gamma_A$ depends intrinsically on the chosen spatial partition, but this relationality encodes a perfectly objective geometric structure in gravitational spacetime. Two observers choosing different partitions will calculate different entropies, not because of subjective ignorance, but because they measure distinct relational geometric invariants.

The second argument concerns the causal relationality revealed by the island formula. This formula generalizes Ryu-Takayanagi according to $S_{\text{fine}}(R) = \min_I \left[ \frac{\text{Area}(\partial I)}{4G\hbar} + S_{\text{bulk}}(R \cup I) \right]$,

<footnote n="7">⁷ R. Penrose, “Singularities and time-asymmetry,” in *General Relativity: An Einstein Centenary Survey*, pp. 581-638 (1979).</footnote>

<footnote n="8">⁸ A. Guth, “Eternal inflation and its implications,” *J. Phys. A* **40**, 6811-6826 (2007).</footnote>

<footnote n="9">⁹ R. Penrose, *Cycles of Time*, Bodley Head (2010).</footnote>

<footnote n="10">¹⁰ A. Ashtekar and P. Singh, “Loop quantum cosmology: A status report,” *Class. Quantum Grav.* **28**, 213001 (2011).</footnote>

<footnote n="11">¹¹ S.W. Hawking, “Breakdown of predictability in gravitational collapse,” *Phys. Rev. D* **14**, 2460-2473 (1976).</footnote>

<footnote n="12">¹² D.N. Page, “Average entropy of a subsystem,” *Phys. Rev. Lett.* **71**, 1291-1294 (1993).</footnote>

<footnote n="13">¹³ G. Penington, “Entanglement wedge reconstruction and the information paradox,” *JHEP* **09**, 002 (2020); A. Almheiri and al., “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” *JHEP* **09**, 063 (2019).</footnote>

<footnote n="14">¹⁴ S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” *Phys. Rev. Lett.* **96**, 181602 (2006).</footnote>
where the optimization encodes the causal connectivity between the outer and inner regions<fnref n="15" />. Before *Page time*, the optimization *selects* the empty island; afterwards, it switches to a non-trivial island, producing a geometric phase transition. This transition demonstrates that *fine-grained* entropy depends essentially on the global causal structure. Two observers with access to causally distinct regions will calculate different entropies, not because of subjective ignorance, but because they measure distinct relational invariants anchored in the objective causal structure.

The third argument is based on the informational emergence proposed by the "*It from Qubit*" program. This program conjectures that spatial geometry emerges from quantum entanglement patterns<fnref n="16" />. If this conjecture holds true, then entropy, which precisely measures entanglement, logically precedes spatial geometry. The relationality of entropy then stems from a fundamental ontological fact: in a universe emerging from quantum information, all quantities necessarily reflect correlations between subsystems, never isolated intrinsic properties.

*Conceptual challenges and testability*

This relational thesis faces three major challenges. The first concerns the distinction between objective relationality and epistemic subjectivity. We defend a distinction inspired by structural relationalism<fnref n="17" />: a quantity is relationally objective if its value depends on invariant physical relations (causality, topology, quantum correlations) rather than arbitrary epistemic conventions (units, coordinates, conventional *coarse-graining*). Gravitational entropy clearly satisfies the first criterion.

The second challenge concerns the compatibility between relationality and objective phenomena such as black hole evaporation. We argue that this compatibility exists by analogy with special relativity: although simultaneity is relative, objective phenomena emerge covariantly. Similarly, although fine-grained entropy is relative, invariant quantities such as the evaporation rate $\dot{M} \propto M^{-2}$ emerge universally. Relationality means structured, non-arbitrary contextuality.

The third challenge concerns empirical testability. We identify three concrete experimental avenues emerging on the horizon for 2025-2035. First, third-generation gravitational interferometry (LISA, Einstein Telescope) could detect quantum deviations in the quasi-normal modes of black holes, testing the unitary corrections predicted by the island formula<fnref n="18" />. Second, quantum simulators implementing the SYK model will allow direct measurement of computational complexity and testing of the Maldacena-Shenker-Stanford chaos bound under controlled conditions. Third, observational cosmology (CMB-S4, LITE-BIRD) will search for signatures of a quantum rebound in the primordial power spectrum, potentially testing one of the predictions of loop quantum gravity<fnref n="19" />.

*Structure of the article*

This article systematically examines four major contemporary debates (2015-2024) that test this relational thesis while revealing its limitations and implications. Part I analyzes the resolution of the information paradox via quantum islands, proposing a precise criterion for experimental testability: if the island formula is correct, late quantum correlations must ex-

<footnote n="15">¹⁵ N. Engelhardt and A.C. Wall, "Quantum extremal surfaces: Holographic entanglement entropy beyond the classical regime," *JHEP* **2015** (01), 073 (2015).</footnote>

<footnote n="16">¹⁶ M. Van Raamsdonk, "Building up spacetime with quantum entanglement," *Gen. Rel. Grav.* **42**, 2323-2329 (2010); B. Swingle, "Entanglement renormalization and holography," *Phys. Rev. D* **86**, 065007 (2012).</footnote>

<footnote n="17">¹⁷ J. Ladyman, "What is structural realism?", *Stud. Hist. Phil. Sci.* **29**, 409-424 (1998); C. Rovelli, "Relational quantum mechanics", *Int. J. Theor. Phys.* **35**, 1637-1678 (1996).</footnote>

<footnote n="18">¹⁸ V. Cardoso, E. Franzin, and P. Pani, "Is the gravitational-wave ringdown a probe of the event horizon?", *Phys. Rev. Lett.* **116**, 171101 (2016). Erratum *Phys. Rev. Lett.* **117**, 089902 (2016).</footnote>

<footnote n="19">¹⁹ A. Ashtekar, T. Pawłowski, and P. Singh, "Quantum nature of the Big Bang," *Phys. Rev. Lett.* **96**, 141301 (2006).</footnote>
hibit an observable phase transition at Page time, detectable by quantum interferometry on 142 simulated radiation in laboratory analogues. Part II argues that the computational complexity $C(t)$, which encodes the post-thermalization dynamics, constitutes an objective physical invariant definable via an intrinsic geometric metric in the space of unitary operators. We predict $C_{max} \sim S_{BH}^{3/2}$, testable in simulations of the SYK model. Part III systematically compares three incompatible unification programs: holographic thermodynamics (Jacobson, Padmanabhan), emergent gravity (Verlinde), and loop quantum gravity. We will construct a table of their distinct observational predictions and argue that they could be related by hidden dualities analogous to AdS/CFT, unifiable via a fundamental quantum tensor formalism. Part IV focuses on entropy-motion duality through the exploration of metric signatures and complex time. Part V finally, evaluates three possible explanations for the initial entropic fine-tuning (multiverse, conformal cycles, quantum bounce) in light of recent cosmological data (DESI 2024, Planck 2018), arguing that loop quantum gravity offers one of the most economical and empirically testable explanations. Our central epistemological contribution articulates a framework that we call quantum entropic relationalism. This position rejects naive realism (entropy as an intrinsic property), instrumentalism (entropy as a mere tool), and Bayesian subjectivism (entropy as an anthropic construction). It argues that entropy is an objective relational structural property encoding real physical relations without being reducible to pre-existing substantial entities. This framework, inspired by Rovelli and Ladyman<sup>20</sup>, resolves the tensions between objectivity and observer dependence by recognizing that objective relational structures can depend on physical contexts without becoming subjective. 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163

# 1. The information paradox and the quantum island formula

The black hole information paradox, formulated by Hawking in 1976, is one of the most profound conceptual crises in contemporary theoretical physics<sup>21</sup>. Its partial resolution via the quantum island formula, developed between 2019 and 2020, transforms our understanding of the relational nature of gravitational entropy. This section first examines the paradox and the constraints imposed by quantum unitarity, then analyzes the island formula and its implications for the causal structure of information, before critically evaluating its limits of validity and empirical testability. 164 165 166 167 168 169 170 171

## 1.1. *The paradox: unitarity versus thermodynamics*

The paradox arises from the confrontation between two seemingly unavoidable principles. Quantum mechanics requires that all physical evolution be unitary, rigorously preserving the information encoded in the quantum state. For an isolated system evolving according to a Hamiltonian $H$, the evolution operator $U(t) = e^{-iHt/\hbar}$ is unitary, implying that the von Neumann entropy of a pure state remains zero: $S[\psi(t)] = -\text{Tr}[\rho(t) \ln \rho(t)] = 0$ for all times $t$. Conversely, Hawking's semi-classical calculation establishes that a black hole of mass $M$ emits thermal radiation at temperature $T_H = \hbar c^3 / (8\pi GMk_B)$, with a Planckian spectrum independent of its formation<sup>22</sup>. This radiation causes gradual evaporation according to $dM/dt \propto -1/M^2$, with a characteristic time $t_{evap} \sim (M/M_\odot)^3 \times 10^{67}$ years. The radiation is described by a mixed thermal state $\rho_{Hawking} = Z^{-1} \exp(-\beta H_{rad})$ with a non-zero entropy proportional to the area of the horizon. 172 173 174 175 176 177 178 179 180 181 182 183

The paradox reaches its apex when considering the complete evolution. An initial pure state $|\psi_{initial}\rangle$ collapses into a black hole, then evaporates completely via thermal radiation. In the semi-classical approximation, the entropy of the radiation increases

<footnote n="20"><sup>20</sup> J. Ladyman and D. Ross, *Every Thing Must Go: Metaphysics Naturalized*, Oxford University Press (2007).</footnote>

<footnote n="21"><sup>21</sup> S.W. Hawking, “Breakdown of predictability in gravitational collapse,” *Phys. Rev. D* **14**, 2460-2473 (1976).</footnote>

<footnote n="22"><sup>22</sup> S.W. Hawking, “Particle creation by black holes,” *Commun. Math. Phys.* **43**, 199-220 (1975).</footnote>
monotonically: $S_{\text{rad}}(t) \approx S_{\text{BH}}(0) - S_{\text{BH}}(t)$ where $S_{\text{BH}}(t)$ decreases with the residual mass. 187 Upon complete evaporation, only high-entropy thermal radiation remains. The overall 188 evolution thus transforms a pure state into a mixed state, violating quantum unitarity. 189 Hawking initially concluded that the information is objectively destroyed, requiring a 190 modification of quantum mechanics in the presence of gravity. This conclusion met with 191 vehement opposition, culminating in 1997 with a famous bet between Hawking and Preskill. 192 Hawking finally conceded the point to Preskill in 2004, without however providing a precise 193 quantitative mechanism$^{23}$. 194

An important quantitative constraint was established by Page in 1993$^{24}$. Consider a bipartite radiation-black hole system evolving unitarily from a pure state. The entanglement entropy of the radiation $S_R = -\text{Tr}_R[\rho_R \ln \rho_R]$ follows a universal evolution curve described as follows: initially, when the radiation contains few quanta compared to the residual black hole, the entanglement increases linearly. At the "Page time" $t_{\text{Page}}$ defined by $S_{\text{BH}}(t_{\text{Page}}) = S_R(t_{\text{Page}})$, the entropy reaches a maximum and then decreases, following $S_R(t) \approx S_{\text{BH}}(t)$ for $t > t_{\text{Page}}$. Upon complete evaporation, the system returns to a pure state: $S_R(t_{\text{evap}}) = 0$. This "Page curve" requires that any unitary theory of quantum gravity must predict a decrease in entropy after the Page time. Hawking's calculation, predicting monotonic growth, clearly violates this constraint. Identifying the physical mechanism responsible for this transition was the central challenge for three decades. 200 201 202 203 204 205

## 1.2. The island formula: geometric transition and unitarity 206

A partial solution emerged in 2019-2020 with the discovery of the "quantum island formula" by Penington and by Almheiri, Engelhardt, Marolf, and Maxfield$^{25}$. This formula generalizes the holographic prescription of Ryu-Takayanagi by incorporating geometric contributions from causally disconnected regions. Consider an asymptotic region $R$ collecting radiation. The fine-grained entropy is given by: 207 208 209 210 211

$$S_{\text{fine}}(R) = \min_{\mathcal{I}} \left[ \frac{\text{Area}(\partial \mathcal{I})}{4G\hbar} + S_{\text{bulk}}(R \cup \mathcal{I}) \right] \quad (1)$$

where the optimization is performed on all possible "islands" $\mathcal{I}$ in the geometric interior, and $S_{\text{bulk}}(R \cup \mathcal{I})$ denotes the von Neumann entropy of the matter fields in $R \cup \mathcal{I}$. The boundary $\partial \mathcal{I}$ contributes to geometric entropy via the generalized Bekenstein-Hawking equation, while $S_{\text{bulk}}$ encodes quantum fluctuations. Physically, this formula describes a dynamic geometric competition. At early times $t \ll t_{\text{Page}}$, optimization selects the empty island $\mathcal{I} = \emptyset$, reproducing the semi-classical result: $S_{\text{fine}}(R) = S_{\text{bulk}}(R) \approx S_{\text{Hawking}}(t)$. Entropy increases linearly with the accumulation of thermal radiation. At Page time, a geometric phase transition occurs. Optimization switches to a configuration where the island $\mathcal{I}$ extends into the interior of the black hole. This island is causally disconnected from $R$ by the horizon, but contributes via the surface term $\text{Area}(\partial \mathcal{I})/(4G\hbar)$. For $t > t_{\text{Page}}$, the dominant contribution comes from this surface: $S_{\text{fine}}(R) \approx S_{\text{BH}}(t)$, which decreases with evaporation. The entropy thus follows Page's curve, guaranteeing unitarity. Almheiri and his collaborators demonstrated that this formula quantitatively reproduces Page's curve in computable models, particularly for eternal black holes in Anti-de Sitter coupled to thermal baths$^{26}$. 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226

<footnote>$^{23}$ S.W. Hawking, "Information loss in black holes," Phys. Rev. D, **72**, 084013 (2005).</footnote>

<footnote>$^{24}$ D.N. Page, "Average entropy of a subsystem," Phys. Rev. Lett. **71**, 1291-1294 (1993).</footnote>

<footnote>$^{25}$ G. Penington, "Entanglement wedge reconstruction and the information paradox," JHEP 09, 002 (2020); A. Almheiri and al., "The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole", JHEP 12, 063 (2019).</footnote>

<footnote>$^{26}$ A. Almheiri and al., "Replica wormholes and the entropy of Hawking radiation," JHEP 05, 013 (2020).</footnote>
The physical interpretation highlights a deeply relational structure of gravitational entropy. Traditionally, the entropy of a region was conceptualized as a local property, depending solely on the degrees of freedom contained in that region. The island formula demonstrates that this locality is illusory in quantum gravity: the *fine-grained* entropy of *R* depends essentially on the global geometry, including causally disconnected regions such as the interior of the black hole. In the AdS/CFT correspondence, the island *I* corresponds to the “quantum entanglement wedge” describing the correlations between radiation and residual gravitational degrees of freedom. The transition at *t*<sub>Page</sub> reflects a topological reorganization of this wedge: before, it encompasses only the exterior; afterwards, it penetrates the horizon and geometrically connects the exterior and interior via emerging “quantum wormholes.” This structure transforms our understanding of the horizon: classically an absolute causal barrier, it becomes quantum permeable in terms of information via entanglement correlations. Hayden and Preskill demonstrated that this informational “leak” is rapid: after throwing a quantum into a black hole, the information reappears after only Δ*t* ~ *M* log *M* (Planck units), much shorter than *t*<sub>evap</sub> ~ *M*<sup>327</sup>.

1.3. *Conceptual limitations and empirical testability*

Despite its success, the island formula faces three major limitations that circumscribe
its validity. First, the semi-classical regime of validity breaks down in the final stages
of evaporation. The derivation assumes a well-defined classical geometry with pertur-
bative quantum corrections. When the residual mass *M*(*t*) approaches the Planck mass
*M*<sub>P</sub> = √ħ*c*/G, the curvature diverges and the quantum fluctuations of the metric become
of order unity. The island formula, based on the geometric identification of extremal sur-
faces ∂*I*, cannot be extrapolated without a complete theory of non-perturbative quantum
gravity. Marolf and Maxfield estimate that corrections become dominant when *S*<sub>BH</sub> ~ *O*(*1*),
suggesting that the last *e*-folds of evaporation escape the semi-classical framework<sup>28</sup>.

Second, generalization to dynamic cosmological spacetime remains problematic. Rigorous derivations rely heavily on the AdS/CFT correspondence, applicable to asymptotically stationary Anti-de Sitter spaces with a conformal boundary. Our observable universe, described by a Friedmann metric with a positive cosmological constant, has neither an AdS asymptote nor a spatial boundary, but a de Sitter cosmological horizon. Generalizing the formula to these realistic spacetimes is a major technical challenge.

Third, direct experimental inaccessibility raises a crucial epistemological question. Stellar-mass astrophysical black holes have evaporation times $t_{\text{evap}} \sim 10^{67}$ years, far exceeding the age of the universe $t_0 \sim 10^{10}$ years. Direct observation of Hawking radiation and its quantum corrections seems impossible with current or foreseeable technology. This inaccessibility calls into question the scientific status of the formula: is it a testable scientific prediction or a metaphysical mathematical construct?

Three indirect testing avenues are nevertheless possible: Quantum simulators can create analogues of black holes in controlled condensed matter systems. Recent experiments with Bose-Einstein condensates have simulated acoustic horizons and measured spectra analogous to that of Hawking<sup>29</sup>. Although profoundly different from gravitational black holes, these systems share universal causal structures and allow for testing unitary predictions in quantum systems. Although profoundly different from gravitational black holes, these systems share universal causal structures and allow unitary predictions to be tested in accessible regimes. Recent proposals aim to create quantum analogues of the SYK model with superconducting qubits, where the Page curve could be measured

<footnote n="27"><sup>27</sup> P. Hayden and J. Preskill, “Black holes as mirrors,” JHEP 09, 120 (2007).</footnote>

<footnote n="28"><sup>28</sup> D. Marolf and H. Maxfield, “Transcending the ensemble,” JHEP 08, 044 (2020).</footnote>

<footnote n="29"><sup>29</sup> J. Steinhauer, "Observation of quantum Hawking radiation", *Nat. Phys.* **12**, 959-965 (2016).</footnote>
directly via quantum state tomography. Third-generation gravitational interferometry (LISA, Einstein Telescope) will achieve sufficient sensitivities to detect quasi-normal modes of black holes with unprecedented precision. If the horizon has a fine microscopic structure, subtle deviations $(\Delta\omega/\omega \sim 10^{-3} \text{ to } 10^{-5})$ from classical predictions will appear in the spectrum of QNMs, potentially resolvable in the next decade<fnref n="30" />. Finally, correlations in the cosmic microwave background could carry signatures of evaporation from microscopic primordial black holes formed in the early universe. The CMB-S4 and LiteBIRD missions, planned for 2030, will achieve sensitivities that will constrain or detect these effects<fnref n="31" />.

## 1.4. Epistemological implications: causal relationality of entropy

Beyond its technical scope, the island formula highlights an elegant epistemological structure concerning the relational nature of entropy. It demonstrates that the fine-grained entropy of a region depends essentially on the overall causal structure of spacetime, including causally inaccessible regions. This relationality has three interconnected dimensions. Spatial relationality implies that entropy depends on the spatial division chosen to partition spacetime: observers using different foliations will calculate geometrically distinct islands and therefore different entropies. Causal relationality implies that entropy depends on the causal structure defined by the observer's light cone: observers at different positions will have access to distinct causal regions, and their entropies will reflect these objective causal differences. Informational relationality indicates that entropy depends on the entanglement correlations between the observed region and its complement, correlations that are objective physical invariants of the global quantum state. This triple relationality transcends the traditional philosophical dichotomy between epistemic subjectivity and ontological objectivity. Gravitational entropy is neither purely subjective (it encodes objective geometric and causal invariants) nor purely objective in the naive sense (it depends intrinsically on the observational context and spatial partitioning). This intermediate position constitutes what we call "objective causal relationality": fundamental physical properties are not intrinsic attributes of substantial entities, but objective relations between subsystems anchored in the causal and informational structure. This perspective aligns with the structural relationalism defended by Rovelli in quantum gravity and Ladyman in the philosophy of science<fnref n="32" />. The island formula possibly foreshadows a notable epistemological change: the absolute notion of entropy as an intrinsic property could be replaced by a relational conception in which entropy would emerge from the overall causal and informational structure of the quantum gravitational universe.

## 2. Entropy of entanglement in AdS/CFT

The Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, conjectured by Maldacena in 1999, is the most precise realization of the holographic principle, establishing an exact duality between quantum gravitational theory in ($d+1$) dimensions and non-gravitational quantum field theory in $d$ dimensions<fnref n="33" />. This correspondence radically transforms our understanding of the link between geometry and quantum information: it suggests that the very structure of geometric spacetime emerges from entanglement correlations in dual quantum theories. This section first examines the Ryu-Takayanagi conjecture linking entropy and geometry, then analyzes the "geometry-from-entanglement" paradigm

<footnote n="30">³⁰ V. Cardoso and al., "Is the Gravitational-Wave Ringdown a Probe of the Event Horizon?", Phys. Rev. Lett. 116, 171101 (2016). Erratum Phys. Rev. Lett. 117, 089902 (2016).</footnote>

<footnote n="31">³¹ B.J. Carr and al., "Constraints on primordial black holes," Rep. Prog. Phys. 84, 116902 (2021).</footnote>

<footnote n="32">³² C. Rovelli, "Relational quantum mechanics," Int. J. Theor. Phys. 35, 1637-1678 (1996); J. Ladyman, "What is structural realism?" Stud. Hist. Phil. Sci. 29, 409-424 (1998).</footnote>

<footnote n="33">³³ J. Maldacena, "The large N limit of superconformal field theories and supergravity," International Journal of Theoretical Physics 38, 1113-1133 (1999).</footnote>
and its implications for the emergence of spacetime, before critically evaluating the limits of validity of the correspondence and its relevance to realistic gravitational physics.

## 2.1. The Ryu-Takayanagi conjecture: quantifying entanglement geometrically

In 2006, Ryu and Takayanagi proposed an important formula quantitatively linking the entanglement entropy of a region in conformal field theory to a geometric quantity in the dual AdS space<fnref n="34" />. Consider a two-dimensional conformal theory CFT₉ defined on the boundary of an Anti-de Sitter space AdS₉₁, in its ground state corresponding to pure AdS space. For a spatial region A on the boundary, its von Neumann entanglement entropy $S_A = -\text{Tr}[\rho_A \ln \rho_A]$, where $\rho_A = \text{Tr}_{\bar{\mathcal{A}}} |\psi\rangle\langle\psi|$ is the reduced density matrix, is given by:

$$S_A = \frac{\text{Area}(\gamma_A)}{4G_N\hbar} \qquad (2)$$

where $\gamma_A$ denotes the extremal surface (minimizing the area) in the geometric interior AdS₉₁ that is homologous to A: partial $\gamma_A = \partial A$ and $\gamma_A$ can be continuously deformed to A without crossing singularities. $G_N$ is Newton's gravitational constant in the bulk.

This formula has several properties that validate its consistency. First, it automatically satisfies the fundamental axioms of entanglement entropy. Strong subadditivity $S_{A\cup B} + S_{A\cap B} \le S_A + S_B$ follows directly from geometric inequalities for minimal surfaces. The monogamy of entanglement $S_A + S_B \ge S_{A\cup B}$ for disjoint regions is verified geometrically via triangulation. Second, it quantitatively reproduces the known results in conformal field theory. For a single interval of length $\ell$ in a CFT₂ with central charge c, the direct calculation of replicas gives $S_A = (c/3) \ln(\ell/\epsilon)$ where $\epsilon$ is an ultraviolet regulator. The Ryu-Takayanagi prescription reproduces this result exactly by identifying $c = 3R/(2G_N)$ where R is the AdS radius of curvature. Third, it generalizes naturally outside of equilibrium. In 2007, Hubeny, Rangamani, and Takayanagi extended the conjecture to time-dependent states by replacing extremal surfaces with extremal 2-co-dimensional surfaces in full spacetime, anchored on given time slices at the boundary<fnref n="35" />.

The conjecture was rigorously proven in the semi-classical regime by several groups between 2013 and 2016. Lewkowycz and Maldacena developed an elegant derivation using the gravitational “replica trick”<fnref n="36" />. The entanglement entropy in the CFT can be calculated via $S_A = \lim_{n\to 1} \frac{\partial}{\partial n} \ln \text{Tr}[\rho_A^n]$. Holographically, $\text{Tr}[\rho_A^n]$ corresponds to the gravitational partition function on a space replicated n times with conical identifications along $\gamma_A$. In the semi-classical limit $G_N \to 0$ (equivalent to $N \to \infty$ for gauge theories), the dominant gravitational action corresponds precisely to the Einstein-Hilbert action evaluated on the extremal surface, reproducing the Ryu-Takayanagi formula. Faulkner and his collaborators demonstrated in 2013 that this derivation generalizes to theories with quantum corrections, establishing the generalized formula including matter contributions<fnref n="37" />:

$$S_A = \frac{\text{Area}(\gamma_A)}{4G_N\hbar} + S_{\text{bulk}}[\gamma_A] \qquad (3)$$

where $S_{\text{bulk}}[\gamma_A]$ represents the von Neumann entropy of quantum matter fields in the region bounded by $\gamma_A$. This generalized formula is precisely the basis of the island formula discussed above.

<footnote n="34">³⁴ S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006).</footnote>

<footnote n="35">³⁵ V.E. Hubeny, M. Rangamani, and T. Takayanagi, “A covariant holographic entanglement entropy proposal,” JHEP 07, 062 (2007).</footnote>

<footnote n="36">³⁶ A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy,” JHEP 08, 090 (2013).</footnote>

<footnote n="37">³⁷ T. Faulkner and al., “Quantum corrections to holographic entanglement entropy,” JHEP 11, 074 (2013).</footnote>
illusion": an effective macroscopic description of an intrinsically non-spatial and non-temporal quantum system.

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## 2.3. *Limits of validity and relevance for realistic gravity*

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Despite its impressive theoretical successes, the AdS/CFT correspondence faces three major limitations that restrict its explanatory power for realistic quantum gravity. First, its dependence on supersymmetry limits its generality. The best-understood examples (notably the original duality between four-dimensional $N = 4$ theory in four dimensions and IIB-type supergravity on $AdS_5 \times S^5$) rely heavily on supersymmetry, which imposes strong algebraic constraints that guarantee consistency and computability. However, the observable universe shows no evidence of exact supersymmetry: no bosonic or fermionic superpartners have been detected up to the TeV scales accessible at the LHC. If supersymmetry is only approximate or absent, the mechanisms ensuring AdS/CFT duality may collapse. Attempts have been made to construct non-supersymmetric dualities (notably via continuous deformations that break supersymmetry), but none have achieved the level of technical control of the supersymmetric examples. This dependence raises the following question: does AdS/CFT represent a universal property of quantum gravity, or is it a mathematical artifact specific to highly symmetrical, unrealistic theories?

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Secondly, the topological and causal gap with our universe is fundamental. Our observable universe has a positive cosmological constant $\Lambda > 0$, corresponding asymptotically to De Sitter spacetime. Anti-De Sitter space has a constant $\Lambda < 0$ with radically different causal properties: AdS space has a temporally accessible spatial boundary at infinity, allowing a quantum theory to be rigorously defined at the boundary. De Sitter space, conversely, has a cosmological horizon with no accessible spatial boundary, making it difficult to define a dual theory. Numerous attempts to construct a “dS/CFT correspondence” have been proposed since 2001, but none has achieved a consensus or precision comparable to that of AdS/CFT. Recently, work by Cotler and Jensen (2021) and Witten (2024) has explored formulations using Euclidean CFTs or “stretched horizon” holographs, but these proposals remain highly speculative and have not produced testable observational predictions<sup>43</sup>. This limitation is crucial: if the AdS/CFT correspondence does not generalize to that of De Sitter, its explanatory power for real cosmology is fundamentally limited.

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Third, the microscopic mechanism of geometric emergence remains unclear. Although the Ryu-Takayanagi conjecture establishes a quantitative correspondence between entropy and geometry, it does not provide an explicit constructive mechanism for reconstructing the bulk metric from the boundary data. What precise mathematical structure in the Hilbert space of the CFT encodes the local geometry? How do localized excitations in the bulk (gravitons, particles) emerge from the operators of the CFT? Substantial progress has been made via “tensor networks”, discrete structures of contracted tensors reproducing the entanglement correlations characteristic of AdS/CFT. In 2012, Swingle demonstrated that the MERA (Multi-scale Entanglement Renormalization Ansatz) tensor network has an emergent hyperbolic geometry analogous to AdS<sup>44</sup>. However, these discrete models reproduce only approximately the continuous limit of AdS, and the precise reconstruction of gravitational dynamics (Einstein’s equations) from tensor rules remains incomplete. Without a constructive microscopic mechanism, AdS/CFT risks remaining an elegant mathematical correspondence rather than a fully explanatory physical theory.

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Despite these limitations, AdS/CFT nevertheless offers relevant indirect testing avenues for real physics. Ultra-relativistic heavy ion collisions create a quark-gluon plasma at

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<footnote n="43">43 E. Witten, "A background-independent algebra in quantum gravity", JHEP 03, 077 (2024).</footnote>

<footnote n="44">44 B. Swingle, "Entanglement renormalization and holography," Phys. Rev. D 86, 065007 (2012).</footnote>
extreme temperatures ($T \sim 10^{12}$ K), behaving like a viscous fluid with a viscosity-entropy ratio $\eta/s$ close to the minimal holographic bound $(4\pi)^{-1}$ predicted by AdS/CFT$^{45}$. Experimental measurements at RHIC and LHC have quantitatively confirmed this prediction with an accuracy of $\sim 20\%$, constituting an interesting indirect test. Strongly correlated condensed matter systems (notably high-temperature superconductors and quantum spin liquids) exhibit properties (transport, thermodynamics) that are qualitatively similar to holographic predictions. Although these systems are not gravitational, they share a similar mathematical structure of strong entanglement, suggesting a universal applicability of holographic methods. Finally, quantum simulators with superconducting qubits or trapped ions are beginning to directly implement tensor networks and structured entangled states. Recent experiments at Google and IBM have created states that reproduce the entanglement structure characteristic of emergent geometries, allowing for experimental testing of "geometry-from-entanglement" conjectures in laboratory-controlled systems$^{46}$.

## 2.4. Implications: informational ontology of spacetime

Beyond its technical applications, AdS/CFT suggests a profound ontological revision of the nature of spacetime. In the classical relativistic conception, spacetime is the fundamental arena in which physical phenomena unfold: a differentiable manifold equipped with a Lorentzian metric, existing independently of matter fields. AdS/CFT radically reverses this ontological hierarchy: spacetime becomes an emergent structure, secondary to quantum correlations in a non-spatial microscopic theory. This perspective aligns with the "structural realism" defended by Ladyman and Ross: fundamental entities are not substances localized in spacetime, but relational structures (entanglement patterns, quantum correlations) from which spacetime itself emerges as a collective approximation$^{47}$.

This emergence has two major epistemological implications. On the one hand, it dissolves the “ground state” problem in quantum gravity: at the fundamental level, no pre-existing geometric structure is required, as geometry itself is dynamically generated by quantum correlations. This dissolution is crucial for unifying gravity and quantum mechanics, as it avoids the circularity of quantizing a (metric) field defined on a geometric structure that is itself dynamic. On the other hand, it suggests an *instrumentalist* conception of classical space-time: smooth geometry appears only within certain limits (large $N$, strong coupling) where microscopic correlations organize themselves collectively. Outside these limits, particularly at the Planck scale, the very notion of spatial locality could become ill-defined, even meaningless. Space-time would thus be analogous to temperature in thermodynamics: a collective emergent property of a system with many degrees of freedom, devoid of meaning for arbitrary microscopic configurations. This radical perspective requires a fundamental change in our understanding of the nature of space-time: it is no longer a fixed, immutable entity, but rather a dynamic, emergent property of a system with many degrees of freedom devoid of meaning for arbitrary microscopic configurations. This radical perspective requires a complete reconceptualization of the nature of physical reality, replacing traditional spatial materialism with a “structural informationism” where quantum information constitutes the fundamental ontology.

<footnote>$^{45}$ P. Kovtun, D.T. Son, and A.O. Starinets, "Viscosity in strongly coupled quantum field theories," Phys. Rev. Lett. 94, 111601 (2005).</footnote>

<footnote>$^{46}$ A. Nahum and al., "Quantum entanglement growth under random unitary dynamics," Phys. Rev. X 7, 031016 (2017).</footnote>

<footnote>$^{47}$ J. Ladyman and D. Ross, Every Thing Must Go: Metaphysics Naturalized, Oxford University Press (2007).</footnote>
### 3. Computational complexity and entropy growth 483

Computational complexity, a central concept in theoretical computer science that quantifies the minimum resources (time, logic gates, memory) required to prepare a quantum state or perform a unitary transformation, has emerged since 2016 as a new fundamental observable in quantum gravity. Unlike entanglement entropy, which rapidly saturates at thermalization, complexity grows linearly over exponentially long timescales, potentially encoding information inaccessible to traditional entropic measures<fnref n="48" />. This section first examines holographic conjectures linking complexity and gravitational geometry, then analyzes the “maximum complexity growth” paradigm and its implications for non-equilibrium thermodynamics, before critically evaluating definitional ambiguities and prospects for empirical testability. 484
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#### 3.1. Holographic conjectures: “complexity = volume” and “complexity = action” 494

Quantum circuit complexity measures the minimum number of elementary gates in a universal set $G = \{U_1, U_2, \dots\}$ needed to prepare a target state $|\psi_{\text{target}}\rangle$ from a reference state $|\psi_{\text{ref}}\rangle$: $C(|\psi_{\text{target}}\rangle) = \min\{n : |\psi_{\text{target}}\rangle = U_{i_1} \cdots U_{i_2} U_{i_1} |\psi_{\text{ref}}\rangle\}$. This definition depends crucially on the choice of the set of gates $G$ and the reference state, ambiguities that we will discuss later. In 2014, Susskind and Stanford proposed that, in the AdS/CFT context, the complexity of a boundary state has a natural geometric dual in the gravitational bulk. Two main conjectures were formulated. 495
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First, the “Complexity = Volume” conjecture (CV), initially proposed by Susskind in 2014, postulates that the circuit complexity of a boundary state $|\psi(\tau)\rangle$ at time $\tau$ is proportional to the maximum spatial volume of a hypersurface in the bulk anchored on the time slice $\tau$ at the boundary<fnref n="49" />: 502
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$$C(\tau) = \frac{V_{\max}(\Sigma_{\tau})}{G_N \ell} \qquad (4)$$

where $\Sigma_{\tau}$ denotes the hypersurface of maximum volume homologous to the slice $\tau$, and $\ell$ is a characteristic scale of the system (typically the AdS radius or thermal length). This conjecture was motivated by the observation that, for an eternal AdS black hole (dual to a maximally entangled double thermofield state), the volume of the region " behind the horizon" region increases linearly with time on exponentially long scales $t \sim e^{S_{BH}}$, a behavior qualitatively similar to the growth of complexity predicted for quantum chaotic systems. 506
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Next, the “Complexity = Action” (C=A) conjecture, proposed by Brown, Roberts, Susskind, Swingle, and Zhao in 2016, instead relates complexity to the gravitational action evaluated in a specific causal region of the bulk<fnref n="50" />: 513
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$$C(\tau) = \frac{S_{\text{WDW}}(W_{\tau})}{\pi \hbar} \qquad (5)$$

where $S_{\text{WDW}}$ denotes the Wheeler-DeWitt gravitational action (Einstein-Hilbert action with appropriate boundary terms) evaluated over the “Wheeler-DeWitt region” $W_{\tau}$, defined as the intersection of the past and future causal cones of the boundary slice $\tau$. This conjecture has the advantage of being manifestly covariant and automatically reproducing the temporal growth properties observed for black holes. 516
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<footnote n="48">⁴⁸ L. Susskind, “Computational complexity and black hole horizons,” Fortsch. Phys. 64, 24-43 (2016).</footnote>

<footnote n="49">⁴⁹ L. Susskind, “Entanglement is not enough,” Fortsch. Phys. 64, 49-71 (2016).</footnote>

<footnote n="50">⁵⁰ A.R. Brown and al., “Holographic complexity equals bulk action?”, Phys. Rev. Lett. 116, 191301 (2016).</footnote>
These conjectures were tested in various contexts where complexity can be calculated independently. For integrable two-dimensional CFTs, between 2006 and 2010, Nielsen and his collaborators developed a geometrization of complexity as a geodesic distance on the space of unitaries equipped with an appropriate Finsler metric<fnref n="51" />. Jefferson and Myers demonstrated in 2017 that, for these systems, the CA conjecture quantitatively reproduces circuit complexity with an appropriate choice of gates and metric, while CV gives qualitatively consistent but quantitatively different results<fnref n="52" />. For tensor network systems that approximately reproduce the AdS/CFT structure, Caputa and his collaborators established in 2018 that the tensor contraction depth (discrete analogue of complexity) is indeed proportional to the emergent geometric volume<fnref n="53" />. These partial validations suggest that the conjectures highlight genuine aspects of the holographic structure, although they do not constitute rigorous mathematical proofs.

## 3.2. Maximal growth and quantum chaos

An important property of gravitational complexity emerges from the dynamic analysis of black holes. For an eternal AdS black hole with mass $M$, temperature $T_H = \beta^{-1}$, and entropy $S_{BH} = A/(4G_N)$, both conjectures predict linear growth of complexity at late times<fnref n="54" />:

$$ \frac{dC}{dt} \approx \frac{2M}{\pi\hbar} = \frac{S_{BH}}{\pi\hbar\beta} \qquad (6) $$

This growth persists on an exponential time scale $t_{scrambling} \sim \beta \ln S_{BH}$ (scrambling time characterizing the complete delocalization of perturbative information), reaching a maximum saturation value $C_{max} \sim e^{S_{BH}}$ corresponding to the complete exploration of exponential Hilbert space. This exponentially prolonged growth contrasts sharply with entanglement entropy, which saturates at thermalization on a scale $\sim \beta: S_{int} \to S_{BH}$.

Susskind proposed in 2018 that this linear growth represents a universal “maximum rate of complexification” for chaotic quantum systems, analogous to a gravitational “Lloyd bound”<fnref n="55" />. The original Lloyd bound, established in quantum computing, states that the maximum rate of quantum operations is bounded by $\dot{C} \le 2E/(\pi\hbar)$, where $E$ is the available energy. Gravitational growth precisely saturates this bound with $E = M$, suggesting that black holes are the most computationally efficient systems in nature: they maximize complexity for a given energy. This enhanced property profoundly links gravity, information, and quantum chaos. The link with quantum chaos was made explicit via “out-of-time-order correlators” (OTOC), quantities that diagnose quantum information scrambling. The OTOC is defined as $F(t) = \langle W^\dagger(t) V^\dagger W(t) V \rangle$ where $W(t) = e^{iHt} We^{-iHt}$ and $V, W$ are local operators. For chaotic systems, $F(t)$ decreases exponentially $F(t) \sim e^{-\lambda_L t}$ with a quantum Lyapunov rate $\lambda_L$ characterizing sensitivity to perturbations. Maldacena, Shenker, and Stanford demonstrated in 2016 that, for holographic systems, the Lyapunov rate saturates the universal bound $\lambda_L \le 2\pi k_B T/\hbar$ (MSS bound), and the growth in complexity is directly related to OTOC decoherence<fnref n="56" />. This connection suggests that complexity quantitatively measures the “depth” of quantum chaos: systems that rapidly reach high complexity exhibit effective information scrambling.

<footnote n="51">51 M.A. Nielsen, “A geometric approach to quantum circuit lower bounds,” *Quantum Inf. Comput.* **6**, 213-262 (2006).</footnote>

<footnote n="52">52 R. Jefferson and R.C. Myers, “Circuit complexity in quantum field theory,” *JHEP* **10**, 107 (2017).</footnote>

<footnote n="53">53 P. Caputa and al., “Anti-de Sitter space from optimization of path integrals in Conformal Field Theories,” *Phys. Rev. Lett.* **119**, 071602 (2017).</footnote>

<footnote n="54">54 D. Stanford and L. Susskind, “Complexity and shock wave geometries,” *Phys. Rev. D* **90**, 126007 (2014).</footnote>

<footnote n="55">55 L. Susskind, “Why do things fall?”, arXiv:1802.01198 (2018).</footnote>

<footnote n="56">56 J. Maldacena, S.H. Shenker, and D. Stanford, “A bound on chaos,” *JHEP* **08**, 106 (2016).</footnote>
This perspective transforms our understanding of non-equilibrium thermodynamics. 560
Traditionally, thermalization is described by entropy reaching its maximum compatible with 561
constraints. However, entropy alone does not describe the fine structure of the thermalized 562
state: many distinct microcanonical states share the same entropy but differ radically in their 563
correlation structure and causal history. Complexity provides a complementary observable 564
characterizing this fine structure. A system can be entropically thermalized ($S \approx S_{max}$) 565
while continuing to become more complex ($C \ll C_{max}$), encoding its historical trajectory 566
in long-range correlations. This distinction is crucial for understanding thermodynamic 567
irreversibility: the increase in entropy characterizes the initial relaxation phase, while the 568
increase in complexity characterizes the prolonged phase of complete scrambling reaching 569
deep statistical equilibrium. Black holes, as paradigmatic gravitational thermodynamic 570
systems, illustrate this temporal distinction between rapid entropic thermalization and 571
prolonged scrambling complexification. 572

### 3.3. *Ambiguities, conceptual challenges, and empirical perspectives* 573

Despite its theoretical potential, holographic complexity faces three major limitations 574
that currently restrict its explanatory and predictive scope. First, the definition of quan- 575
tum complexity suffers from fundamental ambiguities. Circuit complexity is intrinsically 576
dependent on the choice of universal gate set $G$: changing the set quantitatively (and 577
sometimes qualitatively) alters the complexity. For a given state, different sets can yield 578
complexities that differ by polynomial or even exponential factors. Furthermore, the choice 579
of reference state $|\psi_{ref}\rangle$ is arbitrary: should we choose the ground state, the conformal 580
vacuum, or a zero-temperature state? Each choice implicitly encodes physical assumptions 581
about what constitutes microscopic “simplicity” or “triviality.” This definitional ambiguity 582
contrasts with von Neumann entropy, which is defined unambiguously from the density 583
matrix without arbitrary choices. In 2018, Chapman and colleagues explored different 584
definitions (Krylov complexity, Nielsen complexity with various metrics), but no consensus 585
emerged on the “correct” physical definition.<fnref n="57" /> Without a canonical definition, quantitative 586
predictions remain fragile and dependent on the chosen scheme. 587

Second, the ontological status of complexity remains controversial. Is it an objective 588
physical property of the quantum state, or an epistemic property dependent on our com- 589
putational and representational capabilities? Entanglement entropy, although dependent 590
on spatial partitioning, has a clear physical meaning: it quantifies objective quantum 591
correlations between subsystems, measurable via tomography experiments. Complexity, 592
conversely, seems intrinsically linked to our algorithmic description: a state that appears 593
complex in one representation may be simple in another (via a change of basis, hidden 594
symmetries, or compressibility structures). This representational dependence suggests 595
that complexity may be more epistemic than ontological. Aaronson argued in 2016 that in 596
cryptographic contexts, complexity effectively measures an observer's ignorance rather 597
than an intrinsic property<fnref n="58" />. If this interpretation also applies in a holographic context, the 598
physical meaning of the CV and CA conjectures becomes problematic: why would objec- 599
tive geometric quantities (volume, action) correspond to observer-dependent epistemic 600
properties? This ontological tension requires further epistemological clarification. 601

Third, the direct empirical testability of holographic complexity remains extremely 602
limited. Unlike entanglement entropy, which can be measured experimentally using quan- 603
tum interferometry and state tomography techniques, circuit complexity does not have a 604
standard measurement protocol. Measuring the complexity of an arbitrary quantum state 605

<footnote n="57">57 S. Chapman and al., "Complexity and entanglement for thermofield double states", *SciPost Phys.* 6, 034 (2019).</footnote>

<footnote n="58">58 S. Aaronson, "The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes," *Electron. Colloquium Comput. Complex.* TR16 (2016).</footnote>
essentially requires solving the NP-hard problem of finding the minimal circuit, a computationally infeasible task for systems with many qubits. Susskind proposed that the temporal growth of complexity could be accessed via OTOC measurements in quantum simulators, but this connection is rigorously established only for specific holographic models. Nevertheless, three indirect testing avenues are emerging. Quantum entanglement experiments in controlled simulators (trapped ions, superconducting qubits) can measure the growth of Krylov complexity, a quantity algorithmically related to circuit complexity. Zhuang and collaborators performed such measurements on 53-qubit systems in 2019, observing a linear growth qualitatively consistent with holographic predictions<fnref n="59" />. Numerical simulations of QCD on a lattice, modeling quark-gluon plasma, can extract proxies of complexity (thermalization depth, temporal correlation structures) comparable to holographic predictions via gravitational duality. Although these calculations are not yet feasible with quantitative precision, algorithmic developments (tensor methods, hybrid quantum computing) could enable comparisons within the current decade. Finally, observations of gravitational waves from black hole mergers potentially describe indirect signatures of internal complexity. In particular, the “ringdown” (oscillatory phase post-merger) could be sensitive to the complexity structure of the horizon via couplings between quasi-normal modes. In 2019, Brown and his collaborators proposed that subtle deviations in the ringdown spectrum from pure Kerr predictions could signal quantum complexity effects, testable with future LISA and Einstein Telescope detectors<fnref n="60" />.

### 3.4. Implications: towards a thermodynamics of complexity

Beyond the technical challenges, the emergence of complexity as a gravitational observable suggests a conceptual extension of thermodynamics. Traditional thermodynamics (first and second laws) is based on two quantities: energy *E* and entropy *S*, linked by equilibrium relations $dE = TdS - PdV + \mu dN$. This binary structure was extended by Bardeen, Carter, and Hawking in 1973 to black hole thermodynamics, identifying temperature $T_H \propto \kappa$ (surface gravity) and entropy $S_{BH} \propto A$ (horizon area). The CV and CA conjectures suggest that a third fundamental thermodynamic quantity exists: complexity *C*, conjugated to an intensive variable that is still poorly understood.

In 2018, Susskind proposed that this intensive variable be a “complexity pressure” $P_C$, defined by $dC/dt = P_C$ in unitary evolution<fnref n="61" />. This pressure measures the “force” of complexification, reaching its maximum for maximally chaotic systems (black holes). A “first extended law of thermodynamics” would then include $dE = TdS - PdV + \mu dN + P_C dC$, where the complexity term encodes the energy costs of preparing highly structured states. Brown and his collaborators explored this extension in 2019, deriving thermodynamic relations linking temperature, energy, entropy, and complexity for rotating AdS black holes<fnref n="62" />. These relations generalize Smarr’s laws and classical thermodynamic potentials.

This extended thermodynamics has implications for understanding irreversibility. The second law of thermodynamics states that, for an isolated system, $dS/dt \ge 0$ (entropy growth), characterizing relaxation toward thermal equilibrium. A “second law of complexity” could state that $dC/dt \ge 0$ (growth of complexity), characterizing the scrambling and exploration of Hilbert space. Essentially, these two laws operate on radically different time scales: while entropic thermalization occurs on a scale of $\beta$, complete scrambling occurs on a scale of $\beta \ln S$. This temporal separation explains why thermalized systems (entropic

<footnote n="59">⁵⁹ Q. Zhuang and al., “Scrambling and complexity in phase space,” Phys. Rev. A 99, 062334 (2019).</footnote>

<footnote n="60">⁶⁰ A.R. Brown and al., “Quantum gravity in the lab: Teleportation by size and traversable wormholes,” PRX Quantum, 4, 010320 (2023).</footnote>

<footnote n="61">⁶¹ L. Susskind, “Three lectures on complexity and black holes,” arXiv:1810.11563 (2018).</footnote>

<footnote n="62">⁶² A.R. Brown and al., “Complexity, action, and black holes,” Phys. Rev. D 93, 086006 (2016).</footnote>
equilibrium) continue to evolve (complexification): traditional thermodynamic equilibrium corresponds only to a partial equilibrium where macroscopic entropy reaches saturation, while the microscopic structure continues to become more complex until it reaches deep statistical equilibrium, where all memory of the initial conditions is erased. Black holes, as universal thermodynamic attractors, embody this process: they represent the final state of maximum entropy and maximum complexity, simultaneously achieving both forms of equilibrium. This dual perspective fundamentally enriches our understanding of the thermodynamic arrow of time, suggesting that it has two complementary aspects: an entropic arrow (thermal irreversibility) and a complexity arrow (informational irreversibility), both rooted in the causal structure of quantum gravity.

# 4. Entropy-motion duality: metric signatures and complex time

## 4.1. Wick rotation and Euclidean-Lorentzian duality

A little-known but important conceptual thread runs through contemporary theoretical physics: the correspondence between the nature of entropy and the metric signature of spacetime. In the standard formulation, physics operates in Lorentzian signature ($-+, +, +, +$) where time $t$ is distinct from space, allowing for causal propagation and Hamiltonian dynamics. Entropy, on the other hand, seems naturally associated with the Euclidean signature ($+, +, +, +$) obtained by rotating Wick $t \to -i\tau$, transforming real time into pure imaginary time.<fnref n="63" /> This dichotomy suggests a profound duality: the world of energy, characterized by Lorentzian signature and pure real time, governs dynamics and conservation, while the world of information, defined by Euclidean signature and pure imaginary time, encodes the entropic structure and thermal correlations. This conceptual partition between two seemingly disjoint physical regimes emerges naturally as soon as one carefully examines the mathematical formalism underlying fundamental theories.

The connection first appears in the formulation of Feynman's thermal path integral. The canonical partition function $Z = \text{Tr}(e^{-\beta H})$ at temperature $T = 1/(\beta k_B)$ is written as a Euclidean integral with an imaginary period $\beta\hbar$, identifying thermodynamic formalism and Euclidean quantum mechanics.<fnref n="64" /> The thermodynamic entropy $S = -\partial F / \partial T$ emerges naturally from this Euclidean formulation, while the unitary evolution $U(t) = e^{-iHt/\hbar}$ governing real time preserves quantum information via Hamiltonian dynamics. This duality finds geometric expression in the Hartle-Hawking construction for the initial state of the quantum universe, obtained by a Euclidean integral over compact geometries without boundaries, eliminating the initial singularity at the cost of "creating time from space."<fnref n="65" /> The absence of initial temporal boundary conditions—replaced by Euclidean geometric regularity—transforms the cosmological question "why these initial conditions?" into the mathematical constraint "which regular Euclidean geometry?", illustrating how the transition to Euclidean signature reformulates causal problems in informational-geometric terms.

## 4.2. Hawking temperature and KMS periodicity

Hawking radiation crystallizes this duality in an interesting way. A static observer near a Schwarzschild black hole detects a thermal spectrum with temperature $T_H = \hbar c^3 / (8\pi G M k_B)$ even though the global Lorentzian geometry contains no thermal

<footnote n="63">⁶³ G.W. Gibbons and S.W. Hawking, "Action integrals and partition functions in quantum gravity," *Physical Review D* **15** (10), 2752-2756 (1977).</footnote>

<footnote n="64">⁶⁴ R.P. Feynman and A.R. Hibbs, "Quantum Mechanics and Path Integrals," McGraw-Hill (1965); J.J. Halliwell and J.B. Hartle, "Wave functions constructed from an invariant sum over histories satisfy constraints," *Physical Review D* **43** (4), 1170-1194 (1991).</footnote>

<footnote n="65">⁶⁵ J.B. Hartle and S.W. Hawking, "Wave function of the Universe," *Physical Review D* **28** (12), 2960-2975 (1983).</footnote>
matter.<fnref n="66" /> The deep origin lies in a complex analytical structure of the quantum vacuum: the vacuum modes of an accelerated observer in flat Minkowski space (Rindler coordinates) are related to those of an inertial observer by a Wick rotation in the (t, x), transforming the real Lorentzian *boost* into an imaginary Euclidean rotation. This transformation reveals that what one observer considers "vacuum" appears as a "thermal bath" to a differently accelerated observer, a purely kinematic effect encoded in the complex structure of the transformations. This periodicity in imaginary time rigorously defines thermal states in quantum field theory via the KMS (Kubo-Martin-Schwinger) condition: $\langle\phi(t)\phi(0)\rangle = \langle\phi(0)\phi(t + i\beta\hbar)\rangle$, identifying temperature with periodicity in the imaginary time direction.<fnref n="67" /> The Euclidean formalism then reveals an underlying geometry: Schwarzschild's analytically continued spacetime $t \to -i\tau$ becomes a periodic geometry without conical singularities only if the period $\beta = 2\pi/\kappa$, where $\kappa = c^4/(4GM)$, is a surface gravity.<fnref n="68" /> This Euclidean regularity imposes a Hawking temperature as a necessary geometric condition, and not as a thermodynamic property "added" ad hoc to the gravitational system.

The Bekenstein-Hawking entropy $S_{BH} = k_B c^3 A/(4G\hbar)$ is then obtained via the standard thermodynamic relation $S = \beta^2 \partial F / \partial \beta$ applied to the regularized Euclidean action, providing a paradigmatic example of entropy emerging from Euclidean geometric informational structure rather than from counting microstates in Lorentzian dynamic formulation.<fnref n="69" /> This derivation suggests that gravitational entropy is not an ordinary thermodynamic property resulting from our ignorance of microstates, but a fundamental geometric quantity accessible only via analytical continuation to a Euclidean signature. Imaginary time is therefore not a mere computational artifact but an authentic physical dimension encoding thermal-entropic information that does not exist in the real Lorentzian formulation.

### 4.3. Entanglement and geometry: from Euclidean to Lorentzian

The AdS/CFT correspondence extended by Maldacena in 1997 elevates this duality to the rank of a structuring principle of quantum gravity. In its original formulation, the conformal field theory defined on the Euclidean boundary $\mathbb{R}^d$ fully encodes the Euclidean gravitational geometry $\text{AdS}_{d+1}$ in the bulk.<fnref n="70" /> The entanglement entropy $S_A$ of a spatial region $A$ on the boundary — a purely informational quantity defined via the density matrix reduction $\rho_A = \text{Tr}_{\bar{A}}|\psi\rangle\langle\psi|$ then $S_A = -\text{Tr}(\rho_A \log \rho_A)$ — determines the minimum surface area $\gamma_A$ in the Euclidean geometry of the bulk via the Ryu-Takayanagi formula $S_A = \text{Area}(\gamma_A)/(4G\hbar)$.<fnref n="71" /> This geometric relation—Euclidean area is proportional to quantum information correlation—embodies the information-geometry duality in its most explicit form.

However, the Lorentzian continuation required to describe physics with causal time evolution introduces significant conceptual subtleties. The covariant prescription developed by Hubeny, Rangamani, and Takayanagi in 2007 replaces minimal Euclidean surfaces with extremal surfaces $\text{ext}(\gamma_A)$ in Lorentzian geometry, respecting causal constraints but

<footnote n="66">⁶⁶ S.W. Hawking, "Particle creation by black holes", *Communications in Mathematical Physics* **43** (3), 199-220 (1975).</footnote>

<footnote n="67">⁶⁷ R. Haag, N.M. Hugenholtz, and M. Winnink, "On the equilibrium states in quantum statistical mechanics," *Communications in Mathematical Physics* **5** (3), 215-236 (1967); J.J. Bisognano and E.H. Wichmann, "On the duality condition for quantum fields," *Journal of Mathematical Physics* **17** (3), 303-321 (1976).</footnote>

<footnote n="68">⁶⁸ G.W. Gibbons and M.J. Perry, "Black holes and thermal Green functions," *Proceedings of the Royal Society A* **358** (1695), 467-494 (1978).</footnote>

<footnote n="69">⁶⁹ T. Jacobson, "Thermodynamics of spacetime: The Einstein equation of state," *Physical Review Letters* **75** (7), 1260-1263 (1995).</footnote>

<footnote n="70">⁷⁰ E. Witten, "Anti de Sitter space and holography", *Advances in Theoretical and Mathematical Physics* **2** (2), 253-291 (1998).</footnote>

<footnote n="71">⁷¹ S. Ryu and T. Takayanagi, "Holographic derivation of entanglement entropy from AdS/CFT", *Physical Review Letters* **96** (18), 181602 (2006).</footnote>
losing the simple variational definition as a pure geometric minimum.<fnref n="72" /> The transition from Euclidean to Lorentzian thus transforms a mathematically well-posed problem of area minimization into a causally constrained problem of extremization under temporal conditions, reflecting a deep and irreducible tension: entropy, an essentially informational quantity defined at a fixed instant, resists natural incorporation into Lorentzian Hamiltonian dynamics governed by continuous temporal evolution. 730 731 732 733 734 735

This tension culminates in the current debate on “entanglement wedge”: does the entropy $S_A$ calculated via the extremal surface encode the information contained in the causal region of the *bulk* $W(A)$ bounded by null geodesics (dynamic Lorentzian interpretation favoring causality) or in the instantaneous Euclidean “slice” anchored on $\partial A$ (atemporal informational interpretation favoring correlations)? 73 Quantum extremal surfaces generalize the classical prescription by including entropic corrections from the *bulk*, but a conceptual hesitation remains: do these corrections reflect Lorentzian causal dynamics (information propagated by light cones) or an underlying Euclidean structure (geometrically encoded information)? The lack of consensus reveals that the coherent unification of entropy and causality remains an unachieved goal. 736 737 738 739 740 741 742 743 744 745

### 4.4. Complexity and signature: beyond the dichotomy

The holographic complexity program developed by Stanford and Susskind between 2014 and 2020 suggests a partial resolution via the transcendent hybrid concept of pure dichotomy. The computational complexity $C(\rho)$ — the minimum number of unitary quantum gates needed to construct the state $\rho$ from the simple reference — describes the post-thermalization dynamics inaccessible to entropy saturated at the equilibrium value.<fnref n="74" /> Two dual conjectures emerge naturally: complexity proportional to the maximum geodesic volume $C \propto \text{Vol}(\Sigma_{\max})$ anchored on the horizon (CV conjecture) and complexity proportional to the gravitational action integrated over a Wheeler-DeWitt region $C \propto S_{WDW}/\pi\hbar$ bounded by the past and future horizons (CA conjecture). These two propositions intrinsically mix distinct metric signatures in a non-trivial way. The maximum volume $\Sigma_{\max}$ is enclosed by a spatial surface that locally has a Euclidean signature $(+, +, +)$ but is anchored in the dynamic Lorentzian foliation evolving causally according to the time flow. Conversely, the Wheeler-DeWitt action integrates over the Lorentzian causal region bounded by null surfaces (horizons) but measures the “informational volume” analogous to the Euclidean quantity.<fnref n="75" /> This hybridization suggests the necessary overcoming of the pure Euclidean-Lorentzian dichotomy inherited from classical formulations. Complexity, unlike entropy defined at a fixed instant, is intrinsically sequential—it encodes the complete causal history of the construction of the state. Its geometric dual must therefore incorporate a complete Lorentzian causal structure (light cones, horizons, propagation) while measuring an “informational volume” counting degrees of freedom according to Euclidean logic.

Detailed numerical verifications in the SYK (Sachdev-Ye-Kitaev) model confirm this intuition in a quantitatively precise manner: complexity grows linearly $C(t) \sim \lambda t$ on exponentially long time scales $t \sim e^{S_{\text{therm}}}$, well after thermal entropy saturation, capturing the process of information scrambling—an essentially Lorentzian phenomenon governed by

<footnote n="72">72 V.E. Hubeny, M. Rangamani, and T. Takayanagi, “A covariant holographic entanglement entropy proposal,” *JHEP* **2007** (07), 062 (2007).</footnote>

<footnote n="73">73 N. Engelhardt and A.C. Wall, “Quantum extremal surfaces: Holographic entanglement entropy beyond the classical regime,” *JHEP* **2015** (01), 073 (2015).</footnote>

<footnote n="74">74 L. Susskind, “Computational complexity and black hole horizons,” *Fortschritte der Physik* **64** (1), 24–43 (2016); A.R. Brown and al., “Holographic complexity equals bulk action?”, *Physical Review Letters* **116** (19), 191301 (2016).</footnote>

<footnote n="75">75 D. Stanford and L. Susskind, “Complexity and shock wave geometries,” *Physical Review D* **90** (12), 126007 (2014).</footnote>
quantum chaos and causal information propagation.<fnref n="76" /> This prolonged growth, absent from purely Euclidean entropic descriptions, demonstrates the need to incorporate Lorentzian temporal aspects in order to describe the complete informational dynamics of quantum gravitational systems. 771 772 773 774

### 4.5. Implications: towards a mixed signature physics 775

These developments converge towards our conjecture: fundamental physics cannot be formulated either in purely Euclidean signature describing a timeless informational world, or purely in Lorentzian signature describing a causal dynamic world, but requires a mixed signature structure where Euclidean-Lorentzian transitions encode authentic physical degrees of freedom that cannot be eliminated. This thesis finds convergent support in three seemingly independent fields. 776 777 778 779 780 781

Instanton configurations—Euclidean solutions of finite action interpolating between topologically distinct vacua—govern fundamental non-perturbative transitions such as the decay of the metastable vacuum and the nucleation of cosmic bubbles, processes that are completely absent from the classical Lorentzian formulation restricted to continuous Hamiltonian evolution.<fnref n="77" /> These Euclidean “paths,” although appearing “imaginary” in complex formalism, produce real quantitative observable effects via the decay rate $\Gamma \sim e^{-S_{\text{instanton}}/\hbar}$ rate, suggesting that Euclidean configurations possess a hybrid ontology that cannot be reduced to a simple computational artifice. The quantum tunneling effect, already a central phenomenon in non-relativistic quantum mechanics, can be naturally interpreted as a temporary excursion into a Euclidean regime (imaginary time) allowing a barrier crossing that is classically forbidden before returning to real Lorentzian evolution. The formal relationship $\beta \leftrightarrow it$, identifying an inverse temperature and an imaginary time, transcends simple mathematical correspondence to reveal a physical duality: any closed dynamic system evolving unitarily according to a Hamiltonian (Lorentzian temporal description) has a Euclidean “thermal twin” encoding identical correlations via the partition function at effective temperature.<fnref n="78" /> Entropy then appears not as a thermodynamic property “added” by subjective ignorance about microstates, but as an alternative geometric perspective on complete unitary dynamics — the change in metric signature revealing the hidden informational structure present *ab initio* in the Lorentzian description but inaccessible without a complex analytical continuation. This thermofield equivalence suggests that the usual distinction between “reversible unitary dynamics” (Lorentzian) and “irreversible thermal evolution” (Euclidean) reflects the limitation of our conceptual partition rather than a fundamental ontological dichotomy. If spatial geometry does indeed emerge from a quantum entanglement structure according to the Van Raamsdonk program initiated in 2010, and if entanglement is precisely quantized via entropy defined in Euclidean formulation, then “real” Lorentzian spacetime with a causal structure causal structure necessarily emerges from a more fundamental Euclidean informational substrate.<fnref n="79" /> The “sewing” of distinct Euclidean regions via precise thermal identifications (Bisognano’s theorem -Wichmann theorem linking algebraic modularity and the Lorentzian *boost*) gradually reconstructs global Lorentzian causality from purely local informational data. This geometric reconstruction reverses traditional ontological priorities by suggesting that (Euclidean) 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812

<footnote n="76">76 J. Maldacena, S.H. Shenker, and D. Stanford, “A bound on chaos,” JHEP 08, 106 (2016); J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Physical Review D 94 (10), 106002 (2016).</footnote>

<footnote n="77">77 S. Coleman, “The fate of the false vacuum: Semiclassical theory,” Physical Review D 15 (10), 2929-2936 (1977). Erratum Phys. Rev. D 16, 1248 (1977).</footnote>

<footnote n="78">78 R. Kubo, “Statistical-mechanical theory of irreversible processes,” Journal of the Physical Society of Japan 12 (6), 570-586 (1957).</footnote>

<footnote n="79">79 M. Van Raamsdonk, “Building up spacetime with quantum entanglement,” General Relativity and Gravitation 42 (10), 2323-2329 (2010).</footnote>
information-correlation causally precedes (Lorentzian) energy-dynamics, a view that is drastically opposed to the classical intuition favoring the substantial reality of material objects evolving over time.

However, significant conceptual tensions remain unresolved at this stage. Computational complexity, although describing essential Lorentzian aspects such as temporal growth and causal scrambling, still lacks a rigorous mathematical definition independent of the arbitrary choice of elementary gates constituting the computational “alphabet.”<fnref n="80" /> Different sets of gates produce distinct metrics on the state space, complicating identification with the unique invariant geometric quantity. The quantum island formula, a spectacular apparent unification of entropy and geometry that resolves the information paradox, currently applies rigorously only in the semi-classical regime where the Lorentzian-Euclidean dichotomy remains conceptually well defined via standard analytic continuation. With regard to full non-perturbative quantum gravity—whether via spin foam (loop quantum gravity), noncommutative geometries, or M-theory—the very notion of “metric signature” could lose its precise operational meaning, replaced by an abstract algebraic structure from which the Euclidean and Lorentzian signatures would emerge as distinct classical limits corresponding to different asymptotic regimes.<fnref n="81" /> The duality between entropy and motion, a direct physical manifestation of the Euclidean-Lorentzian mathematical duality, is therefore not an isolated technical curiosity but constitutes a window revealing a conceptual architecture of fundamental physical reality. The intuitive opposition between a static informational world characterized by entropy, entanglement, and timeless correlations, and the dynamic energetic world governed by forces, particles, and temporal causality, could be an artifact of our inherited classical formulation, destined to gradually dissolve into a quantum-gravitational synthesis where information and energy, geometry and dynamics, Euclidean and Lorentzian, unify into a mathematical structure that is still largely unimaginable but whose contours are beginning to emerge through these recent developments.

## Conclusion

The systematic exploration of the entropic paradigm highlights a major conceptual transformation in our understanding of fundamental physical reality. This intellectual trajectory, from 19th-century Clausian thermodynamics to contemporary developments in quantum gravity, gradually reveals entropy not as a secondary thermodynamic property or an epistemological artifact of our ignorance, but as a primary organizing structure encoding the constitutive relations of spacetime itself.

Our central contribution articulates the framework of quantum entropic relationalism as an epistemological position that transcends the traditional dichotomies between substantialist realism and instrumentalism. This perspective argues that entropy constitutes an objective relational structural property: it encodes authentic physical relations without being reducible to pre-existing substantial entities or dependent on subjective anthropic constructions. The relational structures thus revealed possess contextual objectivity: their value depends on causal and informational partitioning, but this dependence reflects the physical structure of causality and information rather than arbitrary observational choices. Converging technical results establish this thesis on three complementary pillars. First, the *quantum islands* formula (2019-2020) resolves the black hole information paradox by demonstrating that the fine-grained entropy of a region depends essentially on its overall causal structure: the dynamic emergence of islands inside the horizon reflects the non-local

<footnote n="80">⁸⁰ A.R. Brown and L. Susskind, "Second law of quantum complexity", *Physical Review D* **97** (8), 086015 (2018).</footnote>

<footnote n="81">⁸¹ C. Rovelli and F. Vidotto, *Covariant Loop Quantum Gravity*, Cambridge University Press (2014); J.C. Baez, "Spin foam models," *Classical and Quantum Gravity* **15** (7), 1827-1858 (1998).</footnote>
redistribution of information during Hawking evaporation, preserving unitarity while
respecting semi-classical effective causality. This solution transcends the naive division
between inside and outside, revealing that information is located neither exclusively in the
radiation nor in the residual black hole, but distributed over a relational structure encom-
passing both subsystems and their quantum correlations. Second, the *Complexity=Volume*
and *Complexity=Action* conjectures establish that computational depth (a measure of rela-
tional complexity that quantifies the structural richness of correlations) is identified with
the geometric volume of regions inside the horizon or with gravitational action in the
Wheeler-DeWitt region. This correspondence suggests that temporal evolution is fun-
damentally a process of progressive enrichment of the relational structure: the arrow of
time emerges from the thermodynamic irreversibility of increasing entanglement, with
geometric complexification reflecting informational scrambling that continues well beyond
entropic saturation. The temporal distinction between rapid entropic thermalization (scale
~ β) and prolonged scrambling complexification (scale ~ e^S) describes the complete infor-
mational dynamics that are absent from purely thermodynamic descriptions. Third, the
entanglement-geometry paradigm, crystallized in the Ryu-Takaya-nagi conjecture and its
extensions, establishes that the entanglement entropy of a boundary region is calculated
as the minimal (or extremal) area of surfaces of the *bulk* anchored on that region. This
geometric-informational correspondence highlights that the metric structure emerges from
quantum entanglement patterns: classical geometry appears as a condensed macroscopic
phase from an entangled relational microscopic substrate. Einstein's equations emerge as
thermodynamic equations of state reflecting maximum entropic equilibrium: curvature is
not a deformation of a substantial continuum but a gradient of entanglement density.

Critical analysis nevertheless highlights three major limitations that temper enthusi-
asm. First, the semi-classical validity of the derived formulas collapses at Planck scales
and during the final phases of evaporation, where quantum fluctuations of the metric
become dominant. Second, generalization to realistic cosmological spacetime (notably de
Sitter with accelerated expansion) remains problematic, as most derivations rely heavily
on the AdS/CFT structure, which is difficult to extrapolate. Third, the precise microscopic
mechanisms of "geometric condensation" from entangled states to a smooth classical metric
remain partially opaque, requiring a more detailed understanding of the role of entangle-
ment in the evolution of spacetime. CFT structure that is difficult to extrapolate. Third,
the precise microscopic mechanisms of "geometric condensation" from entangled states
to a smooth classical metric remain partially opaque, potentially requiring a complete
reformulation of quantum dynamics in intrinsically relational terms. Our exploration
of entropy-motion duality via mixed metric signatures deepens this vision. The formal
correspondence β ↔ it between inverse temperature and imaginary time transcends
the computational artifact to reveal a profound physical duality: closed dynamic sys-
tems (temporal Lorentzian description) and their Euclidean "thermal twins" (atemporal
entropic description) encode complementary geometric perspectives on a single infor-
mational reality. Entropy does not appear as a thermodynamic property "added" by
subjective ignorance, but as an alternative geometric perspective on the complete unitary
dynamics: the change in metric signature revealing the underlying informational structure.
Instanton configurations, the quantum tunneling effect, and the growth of complexity
prolonged after entropic saturation converge toward our conjecture: fundamental physics
requires a mixed-signature structure where Euclidean-Lorentzian transitions encode au-
thentic physical degrees of freedom that cannot be eliminated. Emerging experimental
validations, although indirect, qualitatively corroborate this theoretical framework. The
ultrafast thermalization of quark-gluon plasma at RHIC achieves the viscosity-entropy ratio
η/s ≈ (4π)⁻¹ predicted by the holographic bound. Quantum scrambling experiments
on NISQ processors (Google, IBM) directly measure OTOCs and observe the exponen-
tial growth characteristic of quantum chaos with Lyapunov exponents compatible with
the Maldacena-Shenker-Stanford bound $\lambda_L \le 2\pi k_B T/\hbar$. Tensor network simulations
implement states that reproduce the entanglement structure characteristic of emerging
geometries. The next generation of instruments (LISA/Einstein Telescope gravitational
interferometry, CMB-S4/LiteBIRD missions, upgraded EHT telescope) will achieve the
sensitivities required to detect potential signatures of microscopic structure in horizons via
quasi-normal modes, gravitational echoes, or shadow anomalies. On the cosmological front,
our comparative assessment of initial entropy fine-tuning scenarios (multiverse, Penrose
conformal cycles, quantum rebound by concludes that loop quantum gravity offers the
most economical and empirically testable explanation. The quantum bounce replaces the
initial singularity with a previous contraction phase connected via a Planck-scale region
where quantum gravitational repulsion dominates. This resolution naturally preserves
unitarity through the transition, avoiding the fine-tuning problem while producing specific
observational predictions (notably on the primordial tensor spectrum and large-scale CMB
anomalies) that are potentially discriminable with future instrumental precision. specific
observational predictions (notably on the primordial tensor spectrum and large-scale CMB
anomalies) that are potentially discriminable with future instrumental precision.

Beyond the technical aspects, this conceptual revision has profound epistemologi-
cal consequences for our understanding of the nature of the physical world. Traditional
substantialist ontology (positing space-time as an ontologically primitive entity in which
fields and particles evolve) gives way to a processual, relational ontology where infor-
mational relations constitute the primary foundation. Space, time, and local causality
emerge as effective macroscopic structures from a microscopic substrate of non-spatialized
quantum entanglement. This ontological inversion conceptually resolves persistent ten-
sions: the information paradox (non-local information distributed over a global relational
structure), EPR non-locality (remote correlations reveal that spatiality is a derived notion)
, the quantum measurement problem (decoherence emerges from environmental entan-
glement without objective collapse). Research prospects are organized along three main
lines. *Theoretically*, the unification of complementary approaches (AdS/CFT holography,
tensor networks, loop quantum gravity, causal set theories, non-commutative geometry)
remains the priority objective, likely requiring the identification of cross-cutting orga-
nizing principles (principle of geometric emergence from entanglement, the principle of
entropy maximization under causal constraints). Computationally, the development of
dedicated quantum simulators will enable the experimental testing of relational predictions
in controlled systems, gradually building an “experimental science of quantum geometry.”
Observational, optimal use of new-generation instruments will focus on discriminating
signatures: fine structure of horizons via quasi-normal modes, post-merger echoes of
gravitational waves, CMB anomalies compatible with quantum rebound, correlations in
the distribution of primordial black holes. Space-time would thus emerge as an entropic
relational order of a fundamental informational system: not as a pre-existing substance
but as a dynamic network of quantum correlations in perpetual reconfiguration. This
vision, a convergence between quantum gravity, information theory, thermodynamics, and
relational philosophy, outlines the contours of a physics in which entropy, far from being a
measure of subjective ignorance or a secondary thermodynamic property, constitutes the
primary architecture of reality itself.
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