Topics in Algebra of Synchronous Games, Algebraic Graph Identities and Quantum NP-hardness Reductions

We review the correspondence between a synchronous game and its associated game algebra. We slightly develop the work of Helton et al.[HMPS17] by proposing results on algebraic and locally commuting graph identities. Based on the theoretical works on noncommutative Nullstellens\"atze [BWHK23], we build computational tools involving Gr\"obner basis methods and semidefinite programming to check the existence of perfect strategies with specific models. We prove the equivalence between the hereditary and $C$-star models proposed in [HMPS17]. We also extend Ji's reduction $\texttt{3-SAT}\text{-star} \leq_p \texttt{3-Coloring}\text{-star}$ [Ji13] and exhibit another instance of quantum-version NP-hardness reduction $\texttt{3-SAT}\text{-star} \leq_p \texttt{Clique}\text{-star}$.