  We consider statistical methods based on finite samples of locally randomized
measurements in order to certify different degrees of multiparticle
entanglement in intermediate-scale quantum systems. We first introduce
hierarchies of multi-qubit criteria, satisfied by states which are separable
with respect to partitions of different size, involving only second moments of
the underlying probability distribution. Then, we analyze in detail the
statistical error of the estimation in experiments and present several
approaches for estimating the statistical significance based on large deviation
bounds. The latter allows us to characterize the measurement resources required
for the certification of multiparticle correlations, as well as to analyze
given experimental data in detail.
