  We discuss spreading estimates for dynamical systems given by the iteration
of an extended CMV matrix. Using a connection due to
Cantero--Gr\"unbaum--Moral--Vel\'azquez, this enables us to study spreading
rates for quantum walks in one spatial dimension. We prove several general
results which establish quantitative upper and lower bounds on the spreading of
a quantum walk in terms of estimates on a pair of associated matrix cocycles.
To demonstrate the power and utility of these methods, we apply them to several
concrete cases of interest. In the case where the coins are distributed
according to an element of the Fibonacci subshift, we are able to rather
completely describe the dynamics in a particular asymptotic regime. As a
pleasant consequence, this supplies the first concrete example of a quantum
walk with anomalous transport, to the best of our knowldege. We also prove
ballistic transport for a quantum walk whose coins are periodically
distributed.
