  Multipartite entanglement plays an essential role in both quantum information
science and many-body physics. Due to the exponentially large dimension and
complex geometric structure of the state space, the detection of entanglement
in many-body systems is extremely challenging in reality. Conventional means,
like entanglement witness and entropy criterion, either highly depend on the
prior knowledge of the studied systems or the detection capability is
relatively weak. In this work, we propose a framework for designing
multipartite entanglement criteria based on permutation moments, which have an
effective implementation with either the generalized control-SWAP quantum
circuits or the random unitary techniques. These criteria show strong detection
capability in the multi-qubit Ising model with a long-range $XY$ Hamiltonian.
The quantities associated with these criteria have clear physical meaning and
can be used as entanglement quantifiers, with which we show the entanglement
scaling transition in a quantum dynamical phase transition. Furthermore, our
framework can also be generalized to detect the much more complicated
entanglement structure in quantum many-body systems.
