  In this work, building on state-of-the-art quantum Monte Carlo simulations,
we perform systematic finite-size scaling of both entanglement and
participation entropies for long-range Heisenberg chain with unfrustrated
power-law decaying interactions. We find distinctive scaling behaviors for both
quantum entropies in the various regimes explored by tuning the decay exponent
$\alpha$, thus capturing non-trivial features through logarithmic terms, beyond
the case of linear Nambu-Goldstone modes. Our systematic analysis reveals that
the quantum entanglement information, hidden in the scaling of the two studied
entropies, can be obtained to the same level of order parameters and other
usual finite-size observables of quantum many-body lattice models. The analysis
and results obtained here can readily apply to more quantum criticalities in 1D
and 2D systems.
