  We investigate the role of a transverse field on the Ising square
antiferromagnet with first-($J_1$) and second-($J_2$) neighbor interactions.
Using a cluster mean-field approach, we provide a telltale characterization of
the frustration effects on the phase boundaries and entropy accumulation
process emerging from the interplay between quantum and thermal fluctuations.
We found that the paramagnetic (PM) and antiferromagnetic phases are separated
by continuous phase transitions. On the other hand, continuous and
discontinuous phase transitions, as well as tricriticality, are observed in the
phase boundaries between PM and superantiferromagnetic phases. A rich scenario
arises when a discontinuous phase transition occurs in the classical limit
while quantum fluctuations recover criticality. We also find that the entropy
accumulation process predicted to occur at temperatures close to the quantum
critical point can be enhanced by frustration. Our results provide a
description for the phase boundaries and entropy behavior that can help to
identify the ratio $J_2/J_1$ in possible experimental realizations of the
quantum $J_1$-$J_2$ Ising antiferromagnet.
