  The multipartite nonlocality provides deep insights into the fundamental
feature of quantum mechanics and guarantees different degrees of cryptography
security for potential applications in the quantum internet. Verifying
multipartite nonlocal correlations is a difficult task. We propose a unified
approach that encompasses all the quantum characteristics of the multipartite
correlated system beyond from fully separable to biseparable no-signaling
correlations. We offer a straightforward method to verify general systems by
lifting partial nonlocal correlations. This allows to construct a chained Bell
inequality, facilitating the unified verification of hierarchic multipartite
nonlocalities. We finally apply the lifting method to verify the correlations
derived from quantum networks.
