  We study the quantum criticality at finite temperature for three
two-dimensional (2D) $JQ_3$ models using the first principle nonperturbative
quantum Monte Carlo calculations (QMC). In particular, the associated universal
quantities are obtained and their inverse temperature dependence are
investigated. The considered models are known to have quantum phase transitions
from the N\'eel order to the valence bond solid. In addition, these transitions
are shown to be of second order for two of the studied models, with the
remaining one being of first order. Interestingly, we find that the outcomes
obtained in our investigation are consistent with the mentioned scenarios
regarding the nature of the phase transitions of the three investigated models.
Moreover, when the temperature dependence of the studied universal quantities
is considered, a substantial difference between the two models possessing
second order phase transitions and the remaining model is observed. Remarkably,
by using the associated data from both the models that may have continuous
transitions, good data collapses are obtained for a number of the considered
universal quantities. The findings presented here not only provide numerical
evidence to support the results established in the literature regarding the
nature of the phase transitions of these $JQ_3$ models, but also can be
employed as certain promising criterions to distinguish second order phase
transitions from first order ones for the exotic criticalities of the $JQ$-type
models. Finally, based on a comparison between the results calculated here and
the corresponding theoretical predictions, we conclude that a more detailed
analytic calculation is required in order to fully catch the numerical outcomes
determined in our investigation.
