  We consider decision problems on finite sets of hypotheses represented by
pairwise different shift-invariant states on a quantum spin chain. The decision
in favor of one of the hypotheses is based on outputs of generalized
measurements performed on local states on blocks of finite size. We assume
existence of the mean quantum Chernoff distances of any pair of states from the
given set and refer to the minimum of them as the mean generalized quantum
Chernoff distance.
  We establish that this minimum specifies an asymptotic bound on the
exponential decay of the averaged probability of rejecting the true state in
increasing block size, if the mean quantum Chernoff distance of any pair of the
hypothetic states is achievable as an asymptotic error exponent in the
corresponding binary problem. This assumption is in particular fulfiled by
shift-invariant product states (i.i.d. states).
  Further, we provide a constructive proof for the existence of a sequence of
quantum tests in increasing block size, which achieves an asymptotic error
exponent which is equal to the mean generalized quantum Chernoff distance of
the given set of states up to a factor, which depends on the set itself. It can
be arbitrary close to 1 and is not less than $1/m$ for $m$ being the number of
different pairs of states from the set considered.
