  It is well known that entangled quantum states can be nonlocal: the
correlations between local measurements carried out on these states cannot
always be reproduced by local hidden variable models. Svetlichny, followed by
others, showed that multipartite quantum states are even more nonlocal than
bipartite ones in the sense that nonlocal classical models with (super-luminal)
communication between some of the parties cannot reproduce the quantum
correlations. Here we study in detail the kinds of nonlocality present in
quantum states. More precisely we enquire what kinds of classical communication
patterns cannot reproduce quantum correlations. By studying the extremal points
of the space of all multiparty probability distributions, in which all parties
can make one of a pair of measurements each with two possible outcomes, we find
a necessary condition for classical nonlocal models to reproduce the statistics
of all quantum states. This condition extends and generalises work of
Svetlichny and others in which it was shown that a particular class of
classical nonlocal models, the ``separable'' models, cannot reproduce the
statistics of all multiparticle quantum states. Our condition shows that the
nonlocality present in some entangled multiparticle quantum states is much
stronger than previously thought. We also study the sufficiency of our
condition.
