In addition to well-established direct physical linkages between a warming planet and extreme summer weather events such as floods, droughts, heat waves and wildfires are more tenuous but potentially profound linkages tied to altered characteristics of the Northern Hemisphere jet stream and associated wave disturbances. Some of the most significant extreme summer weather events in recent years have been associated with high-amplitude quasi-stationary Rossby waves with zonal wavenumbers 6-8 tied to the phenomenon of Quasi-Resonant Amplification (QRA). Current state-of-the-art (CMIP5) climate models are unable to adequately resolve this mechanism owing to substantial errors in the models in the curvature of the zonal velocity field which is a key quantity controlling the associated wave disturbances. A reliable fingerprint for the occurrence of QRA can, however, be defined in terms of the zonally-averaged surface temperature field. Examining climate observations and historical (CMIP5) climate model simulations, we have demonstrated consistent evidence for an increasing trend in QRA events over the past half century tied to anthropogenic warming. Examining CMIP5 model projections we find that QRA-related extreme weather events are likely to increase by ~50% this century under business-as-usual carbon emissions, but there is considerable variation among climate models. Some predict a near tripling of QRA events by the end of the century, while others predict a potential decrease. Models with amplified Arctic warming yield the most pronounced increase in QRA events. The projections are strongly dependent on assumptions regarding the nature of changes in radiative forcing associated with anthropogenic aerosols over the next century. One implication of our findings is that a reduction in mid-latitude aerosol loading could actually lead to Arctic de-amplification this century, ameliorating potential further increases in persistent extreme weather events. Given the uncertainties, substantial further increases in these events cannot however be ruled out.