  There are several equivalent characterizations of the valuation rank of an
ordered or valued field. In this paper, we extend the theory to the case of an
ordered or valued {\it difference} field (that is, ordered or valued field
endowed with a compatible field automorphism). We introduce the notion of {\it
difference rank}. To treat simultaneously the cases of ordered and valued
fields, we consider quasi-ordered fields. We characterize the difference rank
as the quotient modulo the equivalence relation naturally induced by the
automorphism (which encodes its growth rate). In analogy to the theory of
convex valuations, we prove that any linearly ordered set can be realized as
the difference rank of a maximally valued quasi-ordered difference field. As an
application, we show that for every regular uncountable cardinal $\kappa$ such
that $\kappa= \kappa^{< \kappa}$, there are $2^{\kappa}$ pairwise
non-isomorphic quasi-ordered difference fields of cardinality $\kappa$, but all
isomorphic as quasi-ordered fields.
