  It is by now well-established that there exist non-local games for which the
best entanglement-assisted performance is not better than the best classical
performance. Here we show in contrast that any two-player XOR game, for which
the corresponding Bell inequality is tight, has a quantum advantage. In
geometric terms, this means that any correlation Bell inequality for which the
classical and quantum maximum values coincide, does not define a facet, i.e. a
face of maximum dimension, of the local Bell polytope. Indeed, using
semidefinite programming duality, we prove upper bounds on the dimension of
these faces, bounding it far away from the maximum. In the special case of
non-local computation games, it had been shown before that they are not
facet-defining; our result generalises and improves this. As a by-product of
our analysis, we find a similar upper bound on the dimension of the faces of
the convex body of quantum correlation matrices, showing that (except for the
trivial ones expressing the non-negativity of probability) it does not have
facets.
