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In this work, we study the problem of constructing small sets of orthogonalgeneralized GHZ states that are genuinely nonlocal -- namely, locallyindistinguishable through all bipartitions of the subsystems. We first discussgenuine nonlocality of the canonical generalized GHZ basis. In the multi-qubitcase, using the language of group theory, we show that {\Theta}(2^(N/2)) statesamong the N-qubit GHZ basis suffice to exhibit genuine nonlocality. Such resultcan also be generalized to multi-qudit cases in some situations. Then weconsider generalized GHZ states which are not confined to the canonical form.For any given number of parties N, it is shown that d + 1 genuinely nonlocalgeneralized GHZ states in (C^d)^(\otimes N) exist, whenever the local dimensiond is large enough.

Genuine nonlocality of the generalized GHZ states in multi-partite systems