## 2 Various Questions on Quantum Non-Locality
[72] shows that \(P_{L}=\cos(\theta)\) represents the maximal value when non-maximally entangled states are considered. The decomposition in Equation (4) has been then used for an analysis in terms of non-locality distillation [73]. In this case, when a number of copies of a non-local system is defined, two relevant parameters can be identified: (i) the non-local cost; and (ii) the distillable non-locality. This study put in evidence the existence of non-local boxes, showing what they define as bound non-locality, i.e., whose distillable non-locality is strictly smaller than their non-local cost, analyzing also whether the non-locality contained in such boxes can be activated. In analogy with this, in [74] Bell inequalities are derived exploiting the known link between the Kochen-Specker and Bell theorems, and discovered other measurement bounds allowing a higher degree of non-local correlations than those reported for cases reported in. Several further studies addressed the measurement of non-locality, for instance the study of interconversions among several copies of non-local resources was afforded in [75; 76; 77], the distillation of several copies of a non-local behaviour into a more non-local one in [73; 78; 79], the operations under which non-locality cannot increase used to find a proper definition of genuinely multipartite non-locality [80]. The study about quantification of non-locality naturally leads to the question of the amount of non-locality in quantum mechanics. In this sense, a different approach, that overturns in some sense the perspective on how non-locality can be conceived, is built on reversing the logical order of quantum mechanics [81], by making non-locality an axiom and indeterminism a theorem (QM relies on the opposite). In other words, it is argued that QM is not the most non-local theory (concerning with non-locality correlations) consistent with relativistic causality. This perspective gave the idea to account for non-local \"superquantum\" correlations (known also as PR-box, by [PERSON] and [PERSON]), that interrelate, without superluminal signalling, the outputs of two parties with their inputs A,B in this way:  
\[\sum_{b}P(a,b|A,B)=P(a|A)and\sum_{a}P(a,b|A,B)=P(b|B). \tag{5}\]  
In Ref. [82] the properties of such kind of set of probabilities are detailed in terms of communication complexity. For exemplifying PR-box one can consider the situation when one has the following relation between outputs and inputs:  
\[a\bigoplus b=A\cdot B \tag{6}\]  
where \(\bigoplus\) stands for addition modulus 2. Furthermore, [PERSON]'s and [PERSON]'s marginal distributions are completely random and relativity causality is satisfied (namely [PERSON] and [PERSON] cannot communicate directly by exploiting PR-boxes.) It is worth to mention that these non-local superquantum correlations generate a stronger violation (\(S=4\)) in CHSH inequality with respect to any other quantum correlation, remembering that Tsirelsons bound [54] gives \(S=2\sqrt{2}\) (quantum mechanics) and \(S=2\) in classical theory, while eventually a stronger violation [54; 83] can be achieved with a larger number of measurements (chained inequalities) [84]. Substantially, quantum correlations do not violate locality in a stronger way than as allowed by causality (at least for two dimensional systems, for higher dimensions the situation changes [85], using chained inequalities mentioned above). This raises the question if there is some physical principle limiting the correlations to QM ones.