In general terms Quarter Horses are sprinters meaning they reach their top speed at short distances. A Quarter Horse race is a distance of 440 yds or 1/4 mile. There are many exceptions to the norm though.
A QH can jump, but they do not have the muscular or skeletel make-up to endure competitive show jumping. They don't have the stamina lung capacity that an Arabian has for competitive endurance racing nor the stamina or body build for Cross Country that the Thoroughbred is known to have.
I have done many things with my QH mare, even cross trained her to go in English Pleasure as well as Western. The QH shows do include English Equitation, although I never entered any such shows. She and I did speed events, team cattle penning and lots and lots of trail riding.