  Non-local games are widely studied as a model to investigate the properties
of quantum mechanics as opposed to classical mechanics. In this paper, we
consider a subset of non-local games: symmetric XOR games of $n$ players with
0-1 valued questions. For this class of games, each player receives an input
bit and responds with an output bit without communicating to the other players.
The winning condition only depends on XOR of output bits and is constant w.r.t.
permutation of players.
  We prove that for almost any $n$-player symmetric XOR game the entangled
value of the game is $\Theta (\frac{\sqrt{\ln{n}}}{n^{1/4}})$ adapting an old
result by Salem and Zygmund on the asymptotics of random trigonometric
polynomials. Consequently, we show that the classical-quantum gap is $\Theta
(\sqrt{\ln{n}})$ for almost any symmetric XOR game.
