Reducing the number of questions in nonlocal games

We show how a vector-valued version of Schechtmans empirical method can be used to reduce the number of questions in a nonlocal game G while preserving the quotient β*(G)/β(G) of the quantum over the classical bias. We apply our method to the Khot-Vishnoi game, with exponentially many questions per player, to produce a family of games indexed in n with polynomially many (N ≈ n⁸) questions and n answers per player so that the ratio of the quantum over the classical bias is Ω(n/log² n).