  The relevance of shallow-depth quantum circuits has recently increased,
mainly due to their applicability to near-term devices. In this context, one of
the main goals of quantum circuit complexity is to find problems that can be
solved by quantum shallow circuits but require more computational resources
classically.
  Our first contribution in this work is to prove new separations between
classical and quantum constant-depth circuits. Firstly, we show a separation
between constant-depth quantum circuits with quantum advice
$\mathsf{QNC}^0/\mathsf{qpoly}$, and $\mathsf{AC}^0[p]$, which is the class of
classical constant-depth circuits with unbounded-fan in and $\pmod{p}$ gates.
In addition, we show a separation between $\mathsf{QAC}^0$, which additionally
has Toffoli gates with unbounded control, and $\mathsf{AC}^0[p]$. This
establishes the first such separation for a shallow-depth quantum class that
does not involve quantum fan-out gates.
  Secondly, we consider $\mathsf{QNC}^0$ circuits with infinite-size gate sets.
We show that these circuits, along with (classical or quantum) prime modular
gates, can implement threshold gates, showing that
$\mathsf{QNC}^0[p]=\mathsf{QTC}^0$. Finally, we also show that in the
infinite-size gateset case, these quantum circuit classes for
higher-dimensional Hilbert spaces do not offer any advantage to standard qubit
implementations.
