We consider the problem of testing multiple quantum hypotheses $\left\{ {\rho _1^{ \otimes n},...,\rho _r^{ \otimes n}} \right\}$, 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 minimal average error probability Pe decays exponentially to zero, that is, Pe = exp{–ξn + 0(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 and Szkola's conjecture that ξ = mini≠j C(ρi, ρj). The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C\left( {{\rho _i},{\rho _j}} \right): = \max {}_{0 \leqslant s \leqslant 1}\left\{ { - \log Tr\rho _i^s\rho _j^{1 - s}} \right\}$ has been previously identified as the optimal error exponent for testing two hypotheses, $\rho _i^{ \otimes n}$ versus $\rho _j^{ \otimes n}$. 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 Szkola'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.