  We find a close correspondence between generalized Bell inequalities of a
special kind and certain frustrated spin systems. For example, the
Clauser-Horn-Shimony-Holt inequality corresponds to the frustrated square with
the signs +++- for the nearest neighbor interaction between the spins.
Similarly, the Pearle-Braunstein-Cave inequality corresponds to a frustrated
even ring with the corresponding signs +...+-. Upon this correspondence, the
violation of such inequalities by the entangled singlet state in quantum
mechanics is equivalent to the spin system possessing a classical coplanar
ground state, the energy of which is lower than the Ising ground state's
energy. We propose a scheme which generates new inequalities and give further
examples, the frustrated hexagon with additional diagonal bonds and the
frustrated hypercubes in n=3,4,5 dimensions. Surprisingly, the hypercube in n=4
dimensions yields an inequality which is not violated by the singlet state. We
extend the correspondence to other entangled states and XXZ-models of spin
systems.
