  The multipartite quantum networks feature multiple independent sources, in
contrast to the conventional multipartite Bell experiment involving a single
source. So far, network nonlocality has been explored when each source produces
a two-qubit entangled state. In this work, we demonstrate the network
nonlocality when each party performs a black-box measurement, and the dimension
of the system remains unspecified. In an interesting work, by considering each
source produces two-qubit entangled states in the conventional bilocal
scenario, Gisin \emph{et. al.} in https://doi.org/10.1103/PhysRevA.96.020304]
demonstrated a correspondence between the violations of bipartite
Clauser-Horne-Shimony-Halt inequality and the bilocality inequality. We
introduce a variant of the sum-of-squares approach to reproduce their results
without assuming the dimension of the system. We then generalize the argument
for network nonlocality in star-network topology. Further, we propose a new set
of $n$-locality inequalities in star-network configuration where each of the
$n$ parties performs an arbitrary number of dichotomic measurements and
demonstrate the above correspondence between the quantum violations of the
$n$-locality inequalities and the chained Bell inequalities. A similar
correspondence is demonstrated based on a recently formulated family of
$n$-locality inequalities whose optimal quantum violation cannot be obtained
when each source emits a two-qubit entangled state and requires multiple copies
of two-qubit entangled states. Throughout this paper, each party in the network
performs black-box measurements, and the dimension of the system remains
unspecified.
