Topics in Non-local Games: Synchronous Algebras, Algebraic Graph Identities, and Quantum NP-hardness Reductions

Abstract: We review the correspondence between synchronous games and their associated $*$-algebra. Building upon the work of (Helton et al., New York J. Math. 2017), we propose results on algebraic and locally commuting graph identities. Based on the noncommutative Nullstellens\"atze (Watts, Helton and Klep, Annales Henri Poincar\'e 2023), we build computational tools that check the non-existence of perfect $C^*$ and algebraic strategies of synchronous games using Gr\"obner basis methods and semidefinite programming. We prove the equivalence between the hereditary and $C^*$ models questioned in (Helton et al., New York J. Math. 2017). We also extend the quantum-version NP-hardness reduction $\texttt{3-Coloring}^* \leq_p \texttt{3-SAT}^*$ due to (Ji, arXiv 2013) by exhibiting another instance of such reduction $\texttt{Clique}^* \leq_p \texttt{3-SAT}^*$.