  We propose an ordered set of experimentally accessible conditions for
detecting entanglement in mixed states. The $k$-th condition involves comparing
moments of the partially transposed density operator up to order $k$.
Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki
criterion for detecting entanglement. Our empirical studies highlight that the
first four conditions already detect mixed state entanglement reliably in a
variety of quantum architectures. Exploiting symmetries can help to further
improve their detection capabilities. We also show how to estimate moment
inequalities based on local random measurements of single state copies
(classical shadows) and derive statistically sound confidence intervals as a
function of the number of performed measurements. Our analysis includes the
experimentally relevant situation of drifting sources, i.e. non-identical, but
independent, state copies.
