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We consider statistical methods based on finite samples of locally randomizedmeasurements in order to certify different degrees of multiparticleentanglement in intermediate-scale quantum systems. We first introducehierarchies of multi-qubit criteria, satisfied by states which are separablewith respect to partitions of different size, involving only second moments ofthe underlying probability distribution. Then, we analyze in detail thestatistical error of the estimation in experiments and present severalapproaches for estimating the statistical significance based on large deviationbounds. The latter allows us to characterize the measurement resources requiredfor the certification of multiparticle correlations, as well as to analyzegiven experimental data in detail.

Statistically significant tests of multiparticle quantum correlations based on randomized measurements