  It is difficult to construct a post-inflation QCD axion model that solves the
axion quality problem (and hence the Strong CP problem) without introducing a
cosmological disaster. In a post-inflation axion model, the axion field value
is randomized during the Peccei-Quinn phase transition, and axion domain walls
form at the QCD phase transition. We emphasize that the gauge equivalence of
all minima of the axion potential (i.e., domain wall number one) is
insufficient to solve the cosmological domain wall problem. The axion string on
which a domain wall ends must exist as an individual object (as opposed to a
multi-string state), and it must be produced in the early universe. These
conditions are often not satisfied in concrete models. Post-inflation axion
models also face a potential problem from fractionally charged relics; solving
this problem often leads to low-energy Landau poles for Standard Model gauge
couplings, reintroducing the quality problem. We study several examples,
finding that models that solve the quality problem face cosmological problems,
and vice versa. This is not a no-go theorem; nonetheless, we argue that it is
much more difficult than generally appreciated to find a viable post-inflation
QCD axion model. Successful examples may have a nonstandard cosmological
history (e.g., multiple types of cosmic axion strings of different tensions),
undermining the widespread expectation that the post-inflation QCD axion
scenario predicts a unique mass for axion dark matter.
