  The limited computational power of constant-depth quantum circuits can be
boosted by adapting future gates according to the outcomes of mid-circuit
measurements. We formulate computation of a variety of Boolean functions in the
framework of adaptive measurement-based quantum computation using a cluster
state resource and a classical side-processor that can add bits modulo 2,
so-called $l2$-MBQC. Our adaptive approach overcomes a known challenge that
computing these functions in the nonadaptive setting requires a resource state
that is exponentially large in the size of the computational input. In
particular, we construct adaptive $l2$-MBQC algorithms based on the quantum
signal processing technique that compute the mod-$p$ functions with the best
known scaling in the space-time resources (i.e., qubit count, quantum circuit
depth, classical memory size, and number of calls to the side-processor). As
the subject is diverse and has a long history, the paper includes reviews of
several previously constructed algorithms and recasts them as adaptive
$l2$-MBQCs using cluster state resources. Our results constitute an alternative
proof of an old theorem regarding an oracular separation between the power of
constant-depth quantum circuits and constant-depth classical circuits with
unbounded fan-in NAND and mod-$p$ gates for any prime $p$.
