## 2 Various Questions on Quantum Non-Locality
A partially negative answer was provided in [86], where it was shown that quantum theory is, at the level of correlations among systems, not as special as expected identifying a set of non-signalling correlations (defined in terms of efficiently solvable semi-definite programmes) strictly larger than the quantum correlations one, but satisfies all-but-one (information causality, not yet proved) of the device-independent principles proposed for quantum correlations (non-trivial communication complexity, no advantage for non-local computation, macroscopic locality, and local orthogonality). In a first approach, ref. [87], the problem was faced by a communication complexity point of view [82]. Taking into consideration the case of two parties dealing with a Boolean function \(F(a;b)\), and for which each of the parts has knowledge of only one input (a or b), the interesting problem refers to the minimization in communication resources in order that a part can compute and solve the function. In general, the case in which a bit only is necessary results a trivial matter. For an analysis of nontrivial cases, it can be mentioned that [87] reported the comparison between the bounds for success probability, estimated to be: (i) 90.8% in the case of PR-box operating in probabilistic way; and (ii) 85.4% in the case Tsirelson case (considering that the classical level is fixed at 75%). The consequent partition of communication complexity problems in hard- and easy-solving classes can be understood in terms of partial characterization of non-local correlations by measurements on entangled systems. It is of interest the experimental verification [88; 89; 90; 91] reporting that post-selected measurement results can approach the PR bound, while it is well known that in quantum mechanics the limit is fixed by Tsirelson bound. A study [92] focused on general properties relevant to the whole non-signaling theories (predicting Bell inequalities violation) discerns among the following phenomena, that are a consequence of the no-signaling principle and non-locality: intrinsic randomness, monogamy of correlations (in the sense that these correlations cannot be shared among a indefinite number of parties (e.g., when three parties [PERSON], [PERSON] and [PERSON] share an entangled state and a Bell inequality is maximally violated by [PERSON] and [PERSON], than [PERSON] and [PERSON] result to be totally uncorrelated), and for this reason are different with respect to classical analogue), impossibility of perfect cloning, uncertainty due to the incompatibility of two observables, privacy of correlations (if two honest parties know to share correlations with some degree of monogamy, they can estimate and possibly bound their correlations with a third untrusted party), and bounds in the shareability of some states (shareability represents a natural property in the analysis of the monogamy of correlations: a bipartite probability distribution \(P(a,b|A,B)\) is considered to be m shareable with respect to [PERSON], if there exists an \((\text{m}+1)\)-partite distribution \(P(a,b_{1},...,b_{m}|A,B_{1},...,B_{m})\) that is symmetric with respect to \((b_{1},B_{1})...(b_{m},B_{m})\), with marginals \(P(a,b_{i}|A,B_{i})\) equal to the original distribution \(P(a,b|A,B))\). Moreover, it was discovered that the properties of non-local, non-arbitrarily shareable, and positive secrecy contents result to be equivalent, and this turns out to be valid for any distribution. In general, Bell inequality violation implies that: (i) the considered quantum state must be entangled, (ii) the sets of local quantum measurements performed by one or the other party must be incompatible. A logical question to be asked is whether this link is reversed; [PERSON] demonstrated that entanglement does not imply violation of Bell inequalities. One of the most recent and interesting lines of investigation related with quantum non-locality is the connection with the absence of uncertainty free joint measurements of observables eventually used in Bell inequalities tests.