  The vast amount of computational studies on electrical conduction in
solid-state electrolytes is not mirrored by comparable efforts addressing
thermal conduction, which has been scarcely investigated despite its relevance
to thermal management and (over)heating of batteries. The reason for this lies
in the complexity of the calculations: on one hand, the diffusion of ionic
charge carriers makes lattice methods formally unsuitable, due to the lack of
equilibrium atomic positions needed for normal-mode expansion. On the other
hand, the prohibitive cost of large-scale molecular dynamics (MD) simulations
of heat transport in large systems at ab initio levels has hindered the use of
MD-based methods. In this paper, we leverage recently developed
machine-learning potentials targeting different ab initio functionals (PBEsol,
r$^2$SCAN, PBE0) and a state-of-the-art formulation of the Green-Kubo theory of
heat transport in multicomponent systems to compute the thermal conductivity of
a promising solid-state electrolyte, Li$_3$PS$_4$, in all its polymorphs
($\alpha$, $\beta$, and $\gamma$). By comparing MD estimates with lattice
methods on the low-temperature, nondiffusive $\gamma$-Li$_3$PS$_4$, we
highlight strong anharmonicities and negligible nuclear quantum effects, hence
further justifying MD-based methods even for nondiffusive phases. Finally, for
the ion-conducting $\alpha$ and $\beta$ phases, where the multicomponent
Green-Kubo MD approach is mandatory, our simulations indicate a weak
temperature dependence of the thermal conductivity, a glass-like behavior due
to the effective local disorder characterizing these Li-diffusing phases.
