  Entangled quantum systems can exhibit correlations that cannot be simulated
classically. For historical reasons such correlations are called "Bell
inequality violations." We give two new two-player games with Bell inequality
violations that are stronger, fully explicit, and arguably simpler than earlier
work.
  The first game is based on the Hidden Matching problem of quantum
communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This
game can be won with probability 1 by a strategy using a maximally entangled
state with local dimension $n$ (e.g., $\log n$ EPR-pairs), while we show that
the winning probability of any classical strategy differs from ${1}/{2}$ by at
most $O((\log n)/\sqrt{n})$.
  The second game is based on the integrality gap for Unique Games by Khot and
Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here
$n$-dimensional entanglement allows the game to be won with probability
$1/(\log n)^2$, while the best winning probability without entanglement is
$1/n$. This near-linear ratio is almost optimal, both in terms of the local
dimension of the entangled state, and in terms of the number of possible
outputs of the two players.
