  Characterising the correlations that arise from locally measuring a single
part of a joint 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 the NPA hierarchy, reformulated this question as a polynomial
optimisation problem over noncommutative variables and proposed a convergent
hierarchy of necessary conditions, each testable using semidefinite
programming. More recently, the problem of characterising the quantum network
correlations, which arise when locally measuring several independent quantum
systems distributed in a network, has received considerable interest. Several
generalisations of the NPA hierarchy, such as the Scalar Extension
[Pozas-Kerstjens et al, Phys. Rev. Lett. 123, 140503 (2019)], were introduced
while their converging sets remain unknown. In this work, we introduce a new
hierarchy, prove its equivalence to the Scalar Extension, and characterise its
convergence in the case of the simplest network, the bilocal scenario, and
explore its relations with the known generalisations.
