It depends - originally, the ideal quarter horse was bred to be hot enough to work cows and respond to split-second cues, while still being level-headed and calm enough to be a trusty farm and ranch horse. Over time, I believe they've been bred for the extremes in these and training has a lot to do with it. I wouldn't say the breed is split, necessarily, but you do have them all across the spectrum from the horses that are calmer and make excellent and trusty family horses, to the ones that are champion reiners and cutters that you'd never dream of putting a child or beginner on.
Growing up, my family had two half brothers. One was an extremely hot champion reiner and cutter (we bought him post-career) that you couldn't put a beginner on. At the slightest touch, he responded really quickly and you had to be ready for it. A beginner could easily go off or be unsettled because of the sudden response. However, his half brother has the same stud but is from a much more mellow mare. I won the state 4-H reining AND Western Pleasure competitions on him, but you have to know what you're doing or he'll just stay put. He's not hot at all because and we now use him for lessons because he's so gentle and calm. Never underestimate the value of a lazy horse :P