  We develop criteria to detect three classes of nonlocality that have been
shown by Wiseman et al. [Phys. Rev. Lett. 98, 140402 (2007)] to be
nonequivalent: entanglement, EPR steering, and the failure of local
hidden-variable theories. We use the approach of Cavalcanti et al. [Phys. Rev.
Lett. 99, 210405 (2007)] for continuous variables to develop the nonlocality
criteria for arbitrary spin observables defined on a discrete Hilbert space.
The criteria thus apply to multisite qudits, i.e., systems of fixed dimension
d, and take the form of inequalities. We find that the spin moment inequalities
that test local hidden variables (Bell inequalities) can be violated for
arbitrary d by optimized highly correlated nonmaximally entangled states
provided the number of sites N is high enough. On the other hand, the spin
inequalities for entanglement are violated and thus detect entanglement for
such states, for arbitrary d and N, and with a violation that increases with
N.We show that one of the moment entanglement inequalities can detect the
entanglement of an arbitrary generalized multipartite
Greenberger-Horne-Zeilinger state. Because they involve the natural observables
for atomic systems, the relevant spin-operator correlations should be readily
observable in trapped ultracold atomic gases and ion traps.
