  A bridge trisection of a smooth surface in $S^4$ is a decomposition analogous
to a bridge splitting of a link in $S^3$. The Kirby-Thompson invariant of a
bridge trisection measures its complexity in terms of distances between disc
sets in the pants complex of the trisection surface. We give the first
significant bounds for the Kirby-Thompson invariant of spun knots. In
particular, we show that the Kirby-Thompson invariant of the spun trefoil is
15.
