In this work we study a particular class of Bell inequalities involving only direct equality comparisons of outcomes. In the bipartite case, the scenario can be interpreted as a natural generalization of full-correlator inequalities (xor games) beyond binary outputs. We define the subpolytope of the local polytope corresponding to this scenario and solve it for several bipartite and multipartite scenarios by leveraging some structural properties. In doing so, we obtain thousands of new tight inequalities, many of which are also facets of the standard local polytope. We also define unanimous Bell inequalities, a particular case of the previous class applied to the multipartite setting in which only full-equality events (all outcomes equal) are considered. We show that such inequalities can always be written as deterministic nonlocal games, and we give a simple multipartite unanimous family and prove its local bound. We show that most of these inequalities admit quantum violations, and we also display aspects of their importance for nonlocality. For instance, we identify examples where such inequalities can act as dimension witnesses, outcome witnesses, and witnesses of genuine multipartite nonlocality, as well as being relevant to the Clauser-Horne-Shimony-Holt inequality. These results show that these simple and elegant inequalities by themselves provide a powerful tool for discovering new Bell inequalities and device-independent witnesses.

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• Nonlocality
• Quantum correlations in quantum information
• Quantum information theory