  We give an explicit family of XOR games with O(n)-bit questions requiring 2^n
ebits to play near-optimally. More generally we introduce a new technique for
proving lower bounds on the amount of entanglement required by an XOR game: we
show that near-optimal strategies for an XOR game G correspond to approximate
representations of a certain C^*-algebra associated to G. Our results extend an
earlier theorem of Tsirelson characterising the set of quantum strategies which
implement extremal quantum correlations.
