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Characterising the correlations that arise when locally measuring a singlejoint quantum system is one of the main problems of quantum information theory.The seminal work [M. Navascu\'es et al, NJP 10,7,073013 (2008)], known as theNPA hierarchy, reformulated this question as a polynomial optimisation problemover noncommutative variables and proposed a convergent hierarchy of necessaryconditions, each testable using semidefinite programming. More recently, theproblem of characterising the quantum network correlations, which arise whenlocally measuring several independent quantum systems distributed in a network,has received considerable interest. Several generalisations of the NPAhierarchy, such as the Scalar Extension [Pozas-Kerstjens \emph{et al}, Phys.Rev. Lett. 123, 140503 (2019)], were introduced while their converging setsremain unknown. In this work, we introduce a new hierarchy and characterise itsconvergence in the case of the simplest network, the bilocal scenario, andexplore its relations with the known generalisations. We also provide argumentsagainst a commutator-based definition of the set of quantum correlations innetworks.

Two convergent NPA-like hierarchies for the quantum bilocal scenario