  A blow-up of $n$ copies of a graph $G$ is the graph $\overset{n}\uplus~G$
obtained by replacing every vertex of $G$ by an independent set of size $n$,
where the copies of vertices in $G$ are adjacent in the blow-up if and only if
the vertices adjacent in $G$. Our goal is to investigate the existence of
quantum state transfer on a blow-up graph $\overset{n}\uplus~G$, where the
adjacency matrix is taken to be the time-independent Hamiltonian of the quantum
system represented by $\overset{n}\uplus~G$. In particular, we establish
necessary and sufficient conditions for vertices in a blow-up graph to exhibit
strong cospectrality and various types of high probability quantum transport,
such as periodicity, perfect state transfer (PST) and pretty good state
transfer (PGST). It turns out, if $\overset{n}\uplus~G$ admits PST or PGST,
then one must have $n=2.$ Moreover, if $G$ has an invertible adjacency matrix,
then we show that every vertex in $\overset{2}\uplus~G$ pairs up with a unique
vertex to exhibit strong cospectrality. We then apply our results to determine
infinite families of graphs whose blow-ups admit PST and PGST.
