  Given any finitely aligned higher-rank graph $\Lambda$ and any unital
commutative ring $R$, the Kumjian-Pask algebra $\mathrm{KP}_R(\Lambda)$ is
known as the higher-rank generalization of Leavitt path algebras. After
characterizing simple Kumjian-Pask algebras by L.O. Clark and Y.E.P. Pangalela
(and others), we focus in this article on the purely infinite simple ones.
Briefly, we show that if $\mathrm{KP}_R(\Lambda)$ is simple and every vertex of
$\Lambda$ is reached from a generalized cycle with an entrance, then
$\mathrm{KP}_R(\Lambda)$ is purely infinite. We next prove a standard dichotomy
for simple Kumjian-Pask algebras: in the case that each vertex of $\Lambda$ is
reached only from finitely many vertices and $\mathrm{KP}_R(\Lambda)$ is
simple, then $\mathrm{KP}_R(\Lambda)$ is either purely infinite or locally
matritial. This result covers all unital simple Kumjian-Pask algebras.
