  We show that a generic element of a space of limit-periodic CMV operators has
zero-measure Cantor spectrum. We also prove a Craig--Simon type theorem for the
density of states measure associated with a stochastic family of CMV matrices
and use our construction from the first part to prove that the Craig--Simon
result is optimal in general. We discuss applications of these results to a
quantum walk model where the coins are arranged according to a limit-periodic
sequence. The key ingredient in these results is a new formula which may be
viewed as a relationship between the density of states measure of a CMV matrix
and its Schur function.
