Equivalence between face nonsignaling correlations, full nonlocality, all-versus-nothing proofs, and pseudotelepathy

We show that a quantum correlation $p$ is in a face of the nonsignaling polytope with no local points if and only if $p$ has nonlocal content 1, if and only if $p$ allows for a Greenberger-Horne-Zeilinger-like proof, and if and only if $p$ provides a perfect strategy for a nonlocal game. That is, face nonsignaling (FNS) correlations, full nonlocality (FN), all-versus nothing (AVN) proofs, and pseudotelepathy (PT) are equivalent. This shows that different resources behind a wide variety of fundamental results are in fact the same resource. We demonstrate that quantum correlations with FNS=FN=AVN=PT do not need to maximally violate a tight Bell inequality. We introduce a method for identifying quantum FNS=FN=AVN=PT correlations and use it to prove quantum mechanics does not allow for FNS=FN=AVN=PT neither in the (3,3;3,2) nor in the (3,2;3,4) Bell scenarios. This solves an open problem that, due to the FNS=FN=AVN=PT equivalence, has implications in several fields.