  We present a method for improving measurements of the entanglement R\'enyi
entropies in quantum Monte Carlo simulations by relating them with measurements
of participation R\'enyi entropies. Exploiting the capability of building
improved estimators for the latter allows to obtain very good estimates for
entanglement R\'enyi entropies. When considering a full system instead of a
bipartition, the method can be further ameliorated providing access to the
thermodynamic R\'enyi entropies with high accuracy. We also explore a
recently-proposed method for the reconstruction of the entanglement spectrum
from entanglement R\'enyi entropies and finally show how potential entanglement
Hamiltonians may be tested for their validity using a comparison with thermal
R\'enyi entropies.
