  Recently, W. Slofstra proved that the set of quantum correlations is not
closed. We prove that the set of synchronous quantum correlations is not
closed, which implies his result, by giving an example of a synchronous game
that has a perfect quantum approximate strategy but no perfect quantum
strategy. We also exhibit a graph for which the quantum independence number and
the quantum approximate independence number are different. We prove new
characterisations of synchronous quantum approximate correlations and
synchronous quantum spatial correlations. We solve the synchronous
approximation problem of Dykema and the second author, which yields a new
equivalence of Connes' embedding problem in terms of synchronous correlations.
