Many applications using discrete dynamics employ either q -difference equations or h -difference equations. In this work, we introduce and study the Hyers–Ulam stability (HUS) of a quantum ( q -difference) equation of Euler type. In particular, we show a direct connection between quantum equations of Euler type and h -difference equations of constant step size h with constant coefficients and an arbitrary integer order. For equation orders greater than two, the h -difference results extend first-order and second-order results found in the literature, and the Euler-type q -difference results are completely novel for any order. In many cases, the best HUS constant is found.