Multi-setting Greenberger-Horne-Zeilinger Paradoxes

Greenberger-Horne-Zeilinger (GHZ) paradox provides an all-versus-nothing test for the quantum nonlocality. In all the GHZ paradoxes known so far each observer is allowed to measure only two alternative observables. Here we shall present a general construction for GHZ paradoxes in which each observer measuring more than two observables given that the system is prepared in the $n$-qudit GHZ state. By doing so we are able to construct a multi-setting GHZ paradox for the $n$-qubit GHZ state, with $n$ being arbitrary, that is genuine $n$-partite, i.e., no GHZ paradox exists when restrict to a subset of number of observers for a given set of Mermin observables. Our result fills up the gap of the absence of a genuine GHZ paradox for the GHZ state of an even number of qubits, especially the four-qubit GHZ state as used in GHZ's original proposal.