  We consider sequential hypothesis testing between two quantum states using
adaptive and non-adaptive strategies. In this setting, samples of an unknown
state are requested sequentially and a decision to either continue or to accept
one of the two hypotheses is made after each test. Under the constraint that
the number of samples is bounded, either in expectation or with high
probability, we exhibit adaptive strategies that minimize both types of
misidentification errors. Namely, we show that these errors decrease
exponentially (in the stopping time) with decay rates given by the measured
relative entropies between the two states. Moreover, if we allow joint
measurements on multiple samples, the rates are increased to the respective
quantum relative entropies. We also fully characterize the achievable error
exponents for non-adaptive strategies and provide numerical evidence showing
that adaptive measurements are necessary to achieve our bounds under some
additional assumptions.
