  The axion solution to the strong CP problem is delicately sensitive to
Peccei-Quinn breaking contributions that are misaligned with respect to QCD
instantons. Heavy QCD axion models are appealing because they avoid this
so-called "quality problem". We show that generic realizations of this
framework can be probed by the LIGO-Virgo-KAGRA interferometers, through the
stochastic gravitational wave (GW) signal sourced by the long-lived axionic
string-domain wall network, and by upcoming measurements of the neutron and
proton Electric Dipole Moments. Additionally, we provide predictions for
searches at future GW observatories, which will further explore the parameter
space of heavy QCD axion models.
