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We show that the following statements are equivalent: (i) A quantumcorrelation p is in a face of the nonsignaling polytope that does not containlocal points. (ii) p has local fraction zero; i.e., p has full nonlocality(FN). (iii) p provides an all-versus-nothing (AVN) orGreenberger-Horne-Zeilinger-like proof of nonlocality. (iv) p is a pseudotelepathy (PT) strategy. These connections imply that a long-standing questionposed by Gisin, M\'ethot, and Scarani of whether quantum PT is possible withminimal requirements is fundamental for quantum information, quantumcomputation, and foundations of quantum mechanics, and can be addressed by avariety of strategies. Here, by combining different methods, we show that theanswer is negative: according to quantum mechanics, nature does not allow forFN/AVN/PT in the (3,3;3,2) Bell scenario. Moreover, we show that FN/AVN/PT isalso impossible in (3,2;3,4). We also study (3,3;3,3) and found no example ofFN/AVN/PT. We discuss the implications of these results and furtherapplications of the methods presented.

Impossibility of bipartite full nonlocality, all-versus-nothing proofs, and pseudo-telepathy in small Bell scenarios