  We show that violation of genuine multipartite Bell inequalities can be
obtained with sampled, probabilistic phase space methods. These genuine Bell
violations cannot be replicated if any part of the system is described by a
local hidden variable theory. The Bell violations are simulated
probabilistically using quantum phase-space representations. We treat
mesoscopically large Greenberger-Horne-Zeilinger (GHZ) states having up to 60
qubits, using both a multipartite SU(2) Q-representation and the positive
P-representation. Surprisingly, we find that sampling with phase-space
distributions can be exponentially faster than experiment. This is due to the
classical parallelism inherent in the simulation of quantum measurements using
phase-space methods. Our probabilistic sampling method predicts a contradiction
with local realism of "Schr\"odinger-cat" states that can be realized as a GHZ
spin state, either in ion traps or with photonic qubits. We also present a
quantum simulation of the observed super-decoherence of the ion-trap "cat"
state, using a phenomenological noise model.
