  This work examines the time complexity of quantum search algorithms on
combinatorial $t$-designs with multiple marked elements using the
continuous-time quantum walk. Through a detailed exploration of $t$-designs and
their incidence matrices, we identify a subset of bipartite graphs that are
conducive to success compared to random-walk-based search algorithms. These
graphs have adjacency matrices with eigenvalues and eigenvectors that can be
determined algebraically and are also suitable for analysis in the
multiple-marked vertex scenario. We show that the continuous-time quantum walk
on certain symmetric $t$-designs achieves an optimal running time of
$O(\sqrt{n})$, where $n$ is the number of points and blocks, even when
accounting for an arbitrary number of marked elements. Upon examining two
primary configurations of marked elements distributions, we observe that the
success probability is consistently $o(1)$, but it approaches 1 asymptotically
in certain scenarios.
