We examine both the degree and the structural stability of inflation persistence at different quantiles of the conditional inflation distribution. Previous research focused exclusively on persistence at the conditional mean of the inflation rate. As economic theory provides reasons for inflation persistence to differ across conditional quantiles, this is a potentially severe constraint. Conventional studies of inflation persistence cannot identify changes in persistence at selected quantiles that leave persistence at the median of the distribution unchanged. Based on post-war US data we indeed find robust evidence for a structural break in persistence at all quantiles of the inflation process in the early 1980s. While prior to the 1980s inflation was not mean reverting, quantile autoregression based unit root tests suggest that since the end of the Volcker disinflation the unit root can be rejected at every quantile of the conditional inflation distribution.