We've heard the hype about quantum dots LCD displays years ago, but back then they were still being researched. Now we can finally see quantum dots displays on the market, thanks to Sony and their Triluminos displays that can be found in the Xperia Z Ultra and the Xperia Z1.
QLED vs LCD
The big promise of quantum dots displays (or QLED displays), is that they can offer accurate colors more easily and a bigger gamut of colors compared to regular LCD's. They should also be more efficient power wise. In this regard, it should be a lot more like an OLED display, than an LCD.
LCD's used to be very behind in color gamut, but in the past few years they've reached at least the sRGB color space, and may even surpass it, so that's not bad at all for the good ol' LCD.
QLED vs OLED
QLED might not reach the same level of color contrast and color gamut as OLED, but it should get close enough, to the point where the difference between AMOLED displays and QLED won't be that noticeable, except on paper. Unlike AMOLED, which has a pretty big burn-in problem, because the organic molecules degrade pretty quickly, QLED doesn't have this problem, so that's a pretty big advantage for QLED over OLED displays.
OLED displays also seem very hard to tune, so you can get accurate colors. It took Samsung several generations to even get close to having a pretty accurate display. On the other hand, because the color of the quantum dots can be modified just by increasing or decreasing the dots, and because the colors in them are more pure, QLED displays can be much more easily calibrated to provide the right colors. Unfortunately, it seems even Sony has decided against taking full advantage of this feature, and are calibrating their displays in a way that gives them more contrast instead of very accurate colors, so their displays can also "pop" like Samsung's Super AMOLED.
Bottom line is that these days all three technologies are a lot more similar in terms of quality, because LCD's have improved so much in the past few years, and have solved most of their issues, while OLED's have had some major advantages, but also some major drawbacks, which they've mostly solved.
In theory, QLED seems to be the most promising, and the best of both worlds between LCD and OLED, but it will depend on how many other display manufacturers will adopt it and try to improve it, because Sony has just begun using it in smartphones, and you could say they're still learning how to take advantage of it fully, while Samsung is pushing heavily OLED displays, and the rest of the industry is still invested in LCD's. But at least that means we have 3 different technologies that will compete with each other, and as always, that should help consumers.