In a synchronous nonlocal game, the sets of questions and answers are the same for both players and whenever the players are given the same question, they lose the game, unless they provide the same answer. A key example is the graph $k$-colouring game, where two players are given vertices of a graph and they need to respond with one of the $k$ colours, such that no adjacent vertices have the same colour. Paulsen, Severini, Stahlke, Todorov, and Winter observed that the existence of perfect quantum strategies for these synchronous nonlocal games was intimately tied to the existence of tracial states on a particular $*$-algebra associated with these games. This insight has spurred many results, while also connecting the study of nonlocal games to the rich theory of traces. Although this connection has been very fruitful, many questions remain about the nature of synchronous algebras. In particular, Helton, Meyer, Paulsen, and Satriano showed that there are synchronous nonlocal games with nontrivial synchronous $*$-algebras, but are not $C^*$-algebras. In this talk, I will discuss another interesting example: a synchronous algebra with a representation on $B(H)$, but has no tracial states! The construction is based on the Magic Square of Mermin and Peres and employs the connection between Boolean constraint system (BCS) algebras and synchronous algebras.

This is based on joint work with William Slofstra (https://arxiv.org/abs/2310.07901).