Almost-quantum correlations and their refinements in a tripartite Bell scenario

We study the set of almost-quantum correlations and their refinements in the simplest tripartite Bell scenario where each party is allowed to perform two dichotomic measurements. In contrast to its bipartite counterpart, we find that there already exist facet Bell inequalities that witness almost-quantum correlations beyond quantum theory in this simplest tripartite Bell scenario. Furthermore, we study the relation between the almost-quantum set and the hierarchy of supersets to the quantum set due to Navascu\'es-Pironio-Ac\'n (NPA) Phys. Rev. Lett. 98, 010401 (2007). While the former lies between the first and the third level of the NPA hierarchy, we find that its second level does not contain and is not contained within the almost-quantum set. Finally, we investigate the hierarchy of refinements to the almost-quantum set due to Moroder et al. Phys. Rev. Lett. 111, 030501 (2013), which converges to the set of quantum correlations producible by quantum states having positive partial transposition. This allows us to consider (approximations of) the ``biseparable'' subsets of the almost-quantum set as well as of the quantum set of correlations, and thereby gain further insights into the subtle similarities and differences between the two sets. In addition, they allow us to identify candidate Bell-like inequalities that can serve as device-independent witnesses for genuine tripartite entanglement.