We consider the problem of testing multiple quantum hypotheses {ρ_1^⊗ n,...,ρ_r^⊗ n}, where an arbitrary prior distribution is given and each of the r hypotheses is n copies of a quantum state. It is known that the average error probability P_e decays exponentially to zero, that is, P_e={-ξ n+o(n)}. However, this error exponent ξ is generally unknown, except for the case that r=2. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum

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... and Szkoł a's conjecture that ξ=_i≠ jC(ρ_i,ρ_j). The right-hand side of this equality is called the multiple quantum Chernoff distance, and C(ρ_i,ρ_j):=_0≤ s≤ 1{-Trρ_i^sρ_j^1-s} has been previously identified as the optimal error exponent for testing two hypotheses, ρ_i^⊗ n versus ρ_j^⊗ n. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szkoł a's lower bound. Specialized to the case r=2, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.