  Kirby and Thompson introduced a non-negative integer-valued invariant, called
the Kirby-Thompson invariant, of a $4$-manifold using trisections. In this
paper, we give some lower bounds for the Kirby-Thompson invariant of certain
$4$-manifolds. As an application, we determine the Kirby-Thompson invariant of
the spin of $L(2,1)$, which is the first example of a $4$-manifold with
non-trivial Kirby-Thompson invariant. We also show that there exist
$4$-manifolds with arbitrarily large Kirby-Thompson invariant.
