  We derive conditions in the form of inequalities to detect the genuine
$N$-partite entanglement of $N$ systems. The inequalities are expressed in
terms of variances of spin operators, and can be tested by local spin
measurements performed on the individual systems. Violation of the inequalities
is sufficient (but not necessary) to certify the multipartite entanglement, and
occurs when a type of spin squeezing is created. The inequalities are similar
to those derived for continuous-variable systems, but instead are based on the
Heisenberg spin-uncertainty relation $\Delta J_{x}\Delta J_{y}\geq|\langle
J_{z}\rangle|/2$. We also extend previous work to derive spin-variance
inequalities that certify the full tripartite inseparability or genuine
multi-partite entanglement among systems with fixed spin $J$, as in
Greenberger-Horne-Zeilinger (GHZ) states and W states where $J=1/2$. These
inequalities are derived from the planar spin-uncertainty relation $(\Delta
J_{x})^{2}+(\Delta J_{y})^{2}\geq C_{J}$ where $C_{J}$ is a constant for each
$J$. Finally, it is shown how the inequalities detect multipartite entanglement
based on Stokes operators. We illustrate with experiments that create
entanglement shared among separated atomic ensembles, polarization-entangled
optical modes, and the clouds of atoms of an expanding spin-squeezed
Bose-Einstein condensate. For each example, we give a criterion to certify the
mutual entanglement.
