  We study optimal and nearly-optimal quantum strategies for non-local XOR
games. First, we prove the following general result: for every non-local XOR
game, there exists a set of relations with the properties: (1) a quantum
strategy is optimal for the game if and only if it satisfies the relations, and
(2) a quantum strategy is nearly optimal for the game if and only if it
approximately satisfies the relations. Next, we focus attention on a specific
infinite family of XOR games: the CHSH(n) games. This family generalizes the
well-known CHSH game. We describe the general form of CHSH(n) optimal
strategies. Then, we adapt the concept of intertwining operator from
representation theory and use that to characterize nearly-optimal CHSH(n)
strategies.
