We study the error exponents in a quantum hypothesis testing problem when both the null-hypothesis and the counter-hypothesis can be correlated states of a spin chain. We prove the theorems for the quantum Chernoff and Hoeffding bounds for a class of states that include certain finitely correlated states as well as the global Gibbs states of translation-invariant finite-range interactions [4,5]. Our method is based on reducing the correlated case to an i.i.d.~problem, for which the problems of Chernoff and Hoeffding bounds have recently been solved [1,2,3,6,7].

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