  The concept of randomized measurements on individual particles has proven to
be useful for analyzing quantum systems and is central for methods like shadow
tomography of quantum states. We introduce $\textit{collective}$ randomized
measurements as a tool in quantum information processing. Our idea is to
perform measurements of collective angular momentum on a quantum system and
actively rotate the directions using simultaneous multilateral unitaries. Based
on the moments of the resulting probability distribution, we propose systematic
approaches to characterize quantum entanglement in a
collective-reference-frame-independent manner. First, we show that existing
spin-squeezing inequalities can be accessible in this scenario. Next, we
present an entanglement criterion based on three-body correlations, going
beyond spin-squeezing inequalities with two-body correlations. Finally, we
apply our method to characterize entanglement between spatially-separated two
ensembles.
