  We establish approximate rigidity results for boolean constraint system (BCS)
nonlocal games. In particular, we show that near-perfect quantum strategies are
approximate representations of the corresponding BCS algebra in the little
Frobenius norm. Likewise, for the class of XOR nonlocal games, we show that
near-optimal quantum strategies are approximate representations of the
corresponding *-algebra associated with the game. In both cases, the norm of
the approximate representations is independent of the quantum state employed in
the strategy. We also show that approximate representations of the BCS (resp.
XOR-algebra) are close to near-perfect (resp. near-optimal) quantum strategies
employing a maximally entangled state for the corresponding game. As a
corollary, any near-perfect BCS (resp. near-optimal XOR) quantum strategy is
close to a near-perfect (resp. near-optimal) quantum strategy using a maximally
entangled state. Lastly, we show that every synchronous algebra is *-isomorphic
to a certain BCS algebra, allowing us to apply our results to the class of
synchronous nonlocal games as well.
