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In this work, we consider the following family of two prover one-round games.In the CHSH_q game, two parties are given x,y in F_q uniformly at random, andeach must produce an output a,b in F_q without communicating with the other.The players' objective is to maximize the probability that their outputssatisfy a+b=xy in F_q. This game was introduced by Buhrman and Massar (PRA2005) as a large alphabet generalization of the celebrated CHSH game---which isone of the most well-studied two-prover games in quantum information theory,and which has a large number of applications to quantum cryptography andquantum complexity.  Our main contributions in this paper are the first asymptotic and explicitbounds on the entangled and classical values of CHSH_q, and the realization ofa rather surprising connection between CHSH_q and geometric incidence theory.On the way to these results, we also resolve a problem of Pawlowski and Winterabout pairwise independent Information Causality, which, beside beinginteresting on its own, gives as an application a short proof of our upperbound for the entangled value of CHSH_q.

Information Causality, Szemer\'{e}di-Trotter and Algebraic Variants of CHSH