  We consider the problem of testing multiple quantum hypotheses
$\{\rho_1^{\otimes n},\ldots,\rho_r^{\otimes n}\}$, 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, $P_e=\exp\{-\xi n+o(n)\}$. However, this error
exponent $\xi$ 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 Szko\l a's conjecture that
$\xi=\min_{i\neq j}C(\rho_i,\rho_j)$. The right-hand side of this equality is
called the multiple quantum Chernoff distance, and
$C(\rho_i,\rho_j):=\max_{0\leq s\leq
1}\{-\log\operatorname{Tr}\rho_i^s\rho_j^{1-s}\}$ 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 Szko\l 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.
