  XOR games are the simplest model in which the nonlocal properties of
entanglement manifest themselves. When there are two players, it is well known
that the bias --- the maximum advantage over random play --- of entangled
players can be at most a constant times greater than that of classical players.
Recently, P\'{e}rez-Garc\'{i}a et al. [Comm. Math. Phys. 279 (2), 2008] showed
that no such bound holds when there are three or more players: the advantage of
entangled players over classical players can become unbounded, and scale with
the number of questions in the game. Their proof relies on non-trivial results
from operator space theory, and gives a non-explicit existence proof, leading
to a game with a very large number of questions and only a loose control over
the local dimension of the players' shared entanglement.
  We give a new, simple and explicit (though still probabilistic) construction
of a family of three-player XOR games which achieve a large quantum-classical
gap (QC-gap). This QC-gap is exponentially larger than the one given by
P\'{e}rez-Garc\'{i}a et. al. in terms of the size of the game, achieving a
QC-gap of order $\sqrt{N}$ with $N^2$ questions per player. In terms of the
dimension of the entangled state required, we achieve the same (optimal) QC-gap
of $\sqrt{N}$ for a state of local dimension $N$ per player. Moreover, the
optimal entangled strategy is very simple, involving observables defined by
tensor products of the Pauli matrices.
  Additionally, we give the first upper bound on the maximal QC-gap in terms of
the number of questions per player, showing that our construction is only
quadratically off in that respect. Our results rely on probabilistic estimates
on the norm of random matrices and higher-order tensors which may be of
independent interest.
