number of parties, providing experimentally amenable device-independent inequality constraints along with quantum correlations violating them, thereby certifying that nature's nonlocality must be boundlessly multipartite.

DOI: 10.1103/physrevlett.127.200401

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N\ensuremath{\ge}3

We introduce the class of genuinely local operations and shared randomness multipartite nonlocal correlations, that is, correlations between

N

parties that cannot be obtained from unlimited shared randomness supplemented by any composition of $(N\ensuremath{-}1)$-shared causal generalized probabilistic theory (GPT) resources. We then show that noisy

N

-partite Greenberger-Horne-Zeilinger (GHZ) quantum states as well as the three-partite

W

quantum state can produce such correlations. This proves, if the operational predictions of quantum theory are correct, that nature's nonlocality must be boundlessly multipartite in any causal GPT. We develop a computational method which certifies that a noisy

N=3

GHZ quantum state with fidelity

85%

satisfies this property, making an experimental demonstration of our results within reach. We provide our definition and contrast it with preexisting notions of genuine multipartite nonlocality. This work extends a more compact parallel Letter [X. Coiteux-Roy et al., No Bipartite-Nonlocal Causal Theory Can Explain Nature's Correlations, Phys. Rev. Lett. 127, 200401 (2021)] on the same subject and provides all the required technical proofs.