  We bound separations between the entangled and classical values for several
classes of nonlocal $t$-player games. Our motivating question is whether there
is a family of $t$-player XOR games for which the entangled bias is $1$ but for
which the classical bias goes down to $0$, for fixed $t$. Answering this
question would have important consequences in the study of multi-party
communication complexity, as a positive answer would imply an unbounded
separation between randomized communication complexity with and without
entanglement. Our contribution to answering the question is identifying several
general classes of games for which the classical bias can not go to zero when
the entangled bias stays above a constant threshold. This rules out the
possibility of using these games to answer our motivating question. A
previously studied set of XOR games, known not to give a positive answer to the
question, are those for which there is a quantum strategy that attains value 1
using a so-called Schmidt state. We generalize this class to mod-$m$ games and
show that their classical value is always at least $\frac{1}{m} + \frac{m-1}{m}
t^{1-t}$. Secondly, for free XOR games, in which the input distribution is of
product form, we show $\beta(G) \geq \beta^*(G)^{2^t}$ where $\beta(G)$ and
$\beta^*(G)$ are the classical and entangled biases of the game respectively.
We also introduce so-called line games, an example of which is a slight
modification of the Magic Square game, and show that they can not give a
positive answer to the question either. Finally we look at two-player unique
games and show that if the entangled value is $1-\epsilon$ then the classical
value is at least $1-\mathcal{O}(\sqrt{\epsilon \log k})$ where $k$ is the
number of outputs in the game. Our proofs use semidefinite-programming
techniques, the Gowers inverse theorem and hypergraph norms.

Quantum Physics