the area of the maximum cross-section of the wormhole and of the minimal surface in the throat that connects it to the asymptotic region. This shows that there are probably different notions of complexity that can be useful for answering different questions about the quantum information-theoretic aspects of gravity, and there is still much to be understood.

Another important question concerns the implications of complexity for many-body systems. In order to characterize properties such as scrambling, chaos, and thermalization, extensive use has been made mostly of two type of observables: low-point correlation functions (especially out-of-time-order correlators), and entanglement entropy. Quantum computational complexity captures properties of the quantum state of a system that are more refined than those visible through these observables. This is why it is sensitive to the evolution of the microstates in the ensemble corresponding to a black hole. It is likely that it can also be used to give new insights into the mechanisms underlying the approach to equilibrium and thermalization, and possibly detect new types of phase transitions (see e.g., [29,30,164]).

We should finally point out again that we did not aim at writing a comprehensive review of the subject, therefore we left out many topics that we felt were too advanced for an introduction, such as the thermodynamics and resource theory aspects of complexity [43,46,165,166], the relation with bulk dynamics (in the sense of reconstructing the Einstein equations in the bulk from the complexity of the boundary) [89], alternative conjectures, most notably the one in [96] (sometimes called CV 2.0), complexity in de Sitter space [52,167–169], the evolution of complexity after a quench [77,170], the relation between complexity and chaos [32,171], other notions of complexity such as the *opera-tor complexity* [172,173]. We hope that our readers will be encouraged to delve further into this fascinating subject and contribute to its development.

**Acknowledgements** We gratefully acknowledge discussions with Adam Chapman, Juan Hernandez and Shan-Ming Ruan. We are also grateful to our many collaborators in papers on the subject of complexity in the past few years. The work of SC is supported by the Israel Science Foundation (grant No. 1417/21) and by the German Research Foundation through a German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”. SC acknowledges the support of Carole and Marcus Weinstein through the BGU Presidential Faculty Recruitment Fund.

**Data Availability Statement** This manuscript has no associated data or the data will not be deposited. [Authors' comment: This paper is a review article in theoretical physics. Therefore, no data will be provided.]

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated

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