  We study lower bounds on the optimal error probability in classical coding
over classical-quantum channels at rates below the capacity, commonly termed
quantum sphere-packing bounds. Winter and Dalai have derived such bounds for
classical-quantum channels; however, the exponents in their bounds only
coincide when the channel is classical. In this paper, we show that these two
exponents admit a variational representation and are related by the
Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in
general classical-quantum channels. Second, we establish a sphere-packing bound
for classical-quantum channels, which significantly improves Dalai's prefactor
from the order of subexponential to polynomial. Furthermore, the gap between
the obtained error exponent for constant composition codes and the best known
classical random coding exponent vanishes in the order of $o(\log n / n)$,
indicating our sphere-packing bound is almost exact in the high rate regime.
Finally, for a special class of symmetric classical-quantum channels, we can
completely characterize its optimal error probability without the constant
composition code assumption. The main technical contributions are two converse
Hoeffding bounds for quantum hypothesis testing and the saddle-point properties
of error exponent functions.
