  We consider the multiple hypothesis testing problem for symmetric quantum
state discrimination between r given states \sigma_1,...,\sigma_r. By splitting
up the overall test into multiple binary tests in various ways we obtain a
number of upper bounds on the optimal error probability in terms of the binary
error probabilities. These upper bounds allow us to deduce various bounds on
the asymptotic error rate, for which it has been hypothesised that it is given
by the multi-hypothesis quantum Chernoff bound (or Chernoff divergence)
C(\sigma_1,...,\sigma_r), as recently introduced by Nussbaum and Szko{\l}a in
analogy with Salikhov's classical multi-hypothesis Chernoff bound. This
quantity is defined as the minimum of the pairwise binary Chernoff divergences
min_{j<k}C(\sigma_j,\sigma_k). It was known already that the optimal asymptotic
rate must lie between C/3 and C, and that for certain classes of sets of states
the bound is actually achieved. It was known to be achieved, in particular,
when the state pair that is closest together in Chernoff divergence is more
than 6 times closer than the next closest pair. Our results improve on this in
two ways. Firstly, we show that the optimal asymptotic rate must lie between
C/2 and C. Secondly, we show that the Chernoff bound is already achieved when
the closest state pair is more than 2 times closer than the next closest pair.
We also show that the Chernoff bound is achieved when at least $r-2$ of the
states are pure, improving on a previous result by Nussbaum and Szko{\l}a.
Finally, we indicate a number of potential pathways along which a proof (or
disproof) may eventually be found that the multi-hypothesis quantum Chernoff
bound is always achieved.
