If Wikipedia is to be believed, then quick time events turn ten this year. But Wikipedia would also have us believe that Carrot Top is a comedian and that the world isn't run by fascist alien reptiles who want to steal our water and eat us, so take that with a grain of salt.
Quick time events, or as I like to call them "oh, balls not this crap again", are those little in-game interludes where the game pops up a prompt for you to PUSH A BUTTON, PUSH IT NOW OR LOSE! This is usually done in the middle of a sweet cutscene you're trying to watch.
From a game design standpoint, I can understand the appeal of quick time events. They can provide a mild skill-based check for pretty much anything. We have established gameplay conventions for things like shooting, fighting, and jumping. (And now, farting. Thank you so much, Peter Molyneux.) But if your design document calls for a sequence where the player must whip up a raspberry soufflé while holding off a clan of rival ninja chefs, then you can either invent an entirely new game mechanic and train players how to use it, or you can just throw in a quick time event and go back to writing horrifyingly bad dialog. It's a simple choice. But quick time events aren't real gameplay in the same way that a picture of a hamburger is not food. It might look tasty enough, but it's not going satisfy you.
It should be noted that quick time events are first and foremost a memory test, and a reflex test second. As luck would have it, my first experience with several console systems was with games that used quick time events. If you're familiar with the PlayStation controller then you probably reflexively reach for the top of the gamepad whenever you see a picture of a triangle. But to someone who is new to the system, they might think of that as the "inventory" button, or "enter vehicle" button. If you ask them to hit triangle, their eyes will flick down to the controller. A QTE doesn't allow for that sort of thing, which means the new player has to stop playing the game and slam up against this arbitrary challenge until they can identify all the symbols in 250 milliseconds or less. This wouldn't be bad if this led to them developing an interesting skill, but a QTE has all the depth of playing Simon Says. Because that's what it is. Knowing the buttons by heart is a "skill" that comes naturally over time, and there's no real reason to arrest newcomers until they acquire it.
To me, the point of playing videogames is to give the player some sort of authorship over the world. It's not a lot, but it's usually at least as interesting as, "will I kill this dude using my shotgun or the rocket launcher?" Players have freedom to make tactical and strategic decisions, to decide how and when to use the abilities at their disposal, and decide how aggressive they want to be given how much health they have and how badly they suck at this particular game. There is a gradient of failure (the health bar) that allows them to accumulate and pay off small mistakes without bringing the game to a grinding halt.
But quick time events have none of these features. The player has no control over the events beyond a simple binary pass / fail. Successfully completing a QTE has the same payoff as not pressing the rewind button when you're watching a movie. If you pull it off, you get to see what happens next. And that's it. If you're going to call that "interactive gameplay" then you have to defend the notion that a DVD player and a copy of Forest Gump is a game.