  For the certification and benchmarking of medium-size quantum devices
efficient methods to characterize entanglement are needed. In this context, it
has been shown that locally randomized measurements on a multiparticle quantum
system can be used to obtain valuable information on the so-called moments of
the partially transposed quantum state. This allows to infer some separability
properties of a state, but it is open how to use the given information in an
optimal and systematic manner. We propose two general entanglement detection
methods based on the moments of the partially transposed density matrix. The
first method is based on the Hankel matrices and provides a family of
entanglement criteria, of which the lowest order reduces to the known $p_3$-PPT
criterion proposed in [A. Elben et al., Phys. Rev. Lett. 125, 200501 (2020)].
The second method is optimal and gives necessary and sufficient conditions for
entanglement based on some moments of the partially transposed density matrix.
