Two-component spin uncertainty relations and multipartite entanglement criteria

We obtain a lower bound on the sum of two spin component variances, for any quantum eigenstate of the total spin operator. This gives for Hilbert spaces of fixed spin J a quantum uncertainty relation where one cannot measure all three spin components. The results can be used to derive entanglement criteria for multiple spins J at separated sites, and are useful to detect entanglement where the mean of the spin components at each site is zero.