  While seemingly undesirable, it is not a surprising fact that there are
certain problems for which quantum computers offer no computational advantage
over their respective classical counterparts. Moreover, there are problems for
which there is no `useful' computational advantage possible with the current
quantum hardware. This situation however can be beneficial if we don't want
quantum computers to solve certain problems fast - say problems relevant to
post-quantum cryptography. In such a situation, we would like to have evidence
that it is difficult to solve those problems on quantum computers; but what is
their exact complexity?
  To do so one has to prove lower bounds, but proving unconditional time lower
bounds has never been easy. As a result, resorting to conditional lower bounds
has been quite popular in the classical community and is gaining momentum in
the quantum community. In this paper, by the use of the QSETH framework
[Buhrman-Patro-Speelman 2021], we are able to understand the quantum complexity
of a few natural variants of CNFSAT, such as parity-CNFSAT or counting-CNFSAT,
and also are able to comment on the non-trivial complexity of
approximate-#CNFSAT; both of these have interesting implications about the
complexity of (variations of) lattice problems, strong simulation and hitting
set problem, and more.
  In the process, we explore the QSETH framework in greater detail than was
(required and) discussed in the original paper, thus also serving as a useful
guide on how to effectively use the QSETH framework.
