Abstract
We present the new exact upper bounds on the maximal Bell violation for the generalized N-qubit GHZ state, the N-qudit GHZ state and, in general, for an arbitrary N-partite quantum state, possibly infinite-dimensional. Our results indicate that, for an N-partite quantum state of any Hilbert space dimension, violation of any Bell-type inequality (either on correlation functions or on joint probabilities) with S settings and any number of outcomes at each site cannot exceed (2S-1)^{N-1}.