  Despite multipartite entanglement being a global property of a quantum state,
a number of recent works have made it clear that it can be quantified using
only local measurements. This is appealing because local measurements are the
easiest to implement on current quantum hardware. However, it remains an open
question what protocol one should use in order to minimize the resources
required to estimate multipartite entanglement from local measurements alone.
In this work, we construct and compare several estimators of multipartite
entanglement based solely on the data from local measurements. We first
construct statistical estimators for a broad family of entanglement measures
using local randomized measurement (LRM) data before providing a general
criterion for the construction of such estimators in terms of projective
2-designs. Importantly, this allows us to de-randomize the multipartite
estimation protocol based on LRMs. In particular, we show how local symmetric
informationally complete POVMs enable multipartite entanglement quantification
with only a single measurement setting. For all estimators, we provide both the
classical post-processing cost and rigorous performance guarantees in the form
of analytical upper bounds on the number of measurements needed to estimate the
measures to any desired precision.
