Scribe: Thomas Vidick

In the past lecture we introduced 3-player variants of both the CHSH and Magic Square games. These have the property that, even though from the point of view of classical players the new games are exactly as hard as the 2-player variants (ω(CHSH2) = ω(CHSH3) and ω(MS2) = ω(MS3)), for quantum players the 3-player variants proved much more challenging: we showed ω∗(CHSH3) = ω(CHSH3) < ω∗(CHSH2) and ω(MS3) ≤ ω∗(MS3) < ω∗(MS2) = 1. The intuition behind the three-player variants is that they make it harder for the quantum players to coordinate. In particular, for the case of CHSH the second player, B, is not told if he is playing CHSH with A or with C: even if he shares EPR pairs with each one of them, which pair should he measure? In this lecture we show that a similar transformation applied to the clause-vs-variable game from our first lecture works, in the following sense. Recall that given any instance φ of a k-CSP on alphabet A we defined a game Gφ as follows: 2-player clause-vs-variable game Gφ. Given a constraint satisfaction problem φ = (C1, . . . , Cm) on n variables xi,