  We investigate signatures of topological quantum phase transitions (TQPTs)
between the $Z_2$ quantum spin liquids (QSLs). In two spatial dimensions, $Z_2$
QSLs and their TQPTs are only well defined at zero temperature ($T=0$), and it
is imperative to clarify their observable signatures under nonzero
temperatures. Here, we present the vestiges of TQPTs between $Z_2$ QSLs with
Majorana fermions in terms of thermal Hall conductivity $\kappa_{xy}$ at
nonzero temperatures. The $\kappa_{xy}/T$ shows characteristic temperature
dependences around TQPTs. We argue that an exponential upturn near $T=0$ and
the peak of $\kappa_{xy}/T$ around massive excitation energy are observable
smoking-gun signals of the TQPTs. Quantum critical fan-shape temperature
dependences are uncovered across TQPTs. We also perform the parton mean-field
analysis on a modified Kitaev model with next-nearest neighbor interactions
finding TQPTs between the phases with different Chern numbers and their
vestiges self-consistently. We discuss the implication of our results to the
recent experiments in $\alpha$-RuCl$_3$.
