  We investigate the List $H$-Coloring problem, the generalization of graph
coloring that asks whether an input graph $G$ admits a homomorphism to the
undirected graph $H$ (possibly with loops), such that each vertex $v \in V(G)$
is mapped to a vertex on its list $L(v) \subseteq V(H)$. An important result by
Feder, Hell, and Huang [JGT 2003] states that List $H$-Coloring is
polynomial-time solvable if $H$ is a so-called bi-arc graph, and NP-complete
otherwise. We investigate the NP-complete cases of the problem from the
perspective of polynomial-time sparsification: can an $n$-vertex instance be
efficiently reduced to an equivalent instance of bitsize $O(n^{2-\varepsilon})$
for some $\varepsilon > 0$? We prove that if $H$ is not a bi-arc graph, then
List $H$-Coloring does not admit such a sparsification algorithm unless $NP
\subseteq coNP/poly$. Our proofs combine techniques from kernelization lower
bounds with a study of the structure of graphs $H$ which are not bi-arc graphs.
