  In this work, we obtain a microscopic derivation of Gibbs measures for the
focusing quintic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{T}$ from
many-body quantum Gibbs states. On the quantum many-body level, the quintic
nonlinearity corresponds to a three-body interaction. This is a continuation of
our previous work. In the aforementioned work, we studied the cubic problem,
which corresponds to a two-body interaction on the quantum many-body level.
  In our setup, we truncate the mass of the classical free field in the
classical setting and the rescaled particle number in the quantum setting. Our
methods are based on a perturbative expansion previously developed in the work
of Fr\"{o}hlich, Knowles, Schlein, and the second author. We prove results both
in the time-independent and time-dependent setting. This is the first such
known result in the three-body regime. Furthermore, this gives the first
microscopic derivation of time-dependent correlation functions for Gibbs
measures corresponding to the quintic NLS, as studied in the work of Bourgain.
