N-particle nonclassicality without N-particle correlations

Most of known multipartite Bell inequalities involve correlation functions for all subsystems. They are useless for entangled states without such correlations. We give a method of derivation of families of Bell inequalities for N parties, which involve, e. g., only (N - 1)-partite correlations, but still are able to detect proper N-partite entanglement. We present an inequality which reveals five-partite entanglement despite using only four-partite correlations. Classes of inequalities introduced here can be put into a handy form of a single nonlinear inequality. An example is given of an N-qubit state, which strongly violates such an inequality, despite having no N-qubit correlations. This surprising property might be of potential value for quantum information tasks.