DisplayMate, a display testing company, has released a much awaited report (link) comparing the screens of:

• Apple iPad Mini
• Google Nexus 7 (2013)
• Amazon Kindle HDX

The bottom line is that Apple's iPad mini has the worse display using "out of date" technology.

"Amazon and Google, have significantly leapfrogged Apple by introducing Tablet displays using LTPS (in the Kindle Fire HDX 8.9 and the new Nexus 7), and they are significantly outperforming the IGZO and a-Si displays in the current iPads. Apple was once the leader in mobile displays, unfortunately it has fallen way behind in both Tablets and Smartphones."

DisplayMate explains that the Amazon HDX has the best display using a technology called Quantum Dots:

"Quantum Dots are almost magical because they use Quantum Physics to produce highly saturated primary colors for LCDs that are similar to those produced by OLED displays. They not only significantly increase the size of the Color Gamut by 40-50 percent but also improve the power efficiency by an additional 15-20 percent. Instead of using White LEDs (which have yellow phosphors) that produce a broad light spectrum that makes it hard to efficiently produce saturated colors, Quantum Dots directly convert the light from Blue LEDs into highly saturated primary colors for LCDs. "

They further explain that:

"The new iPad mini with Retina Display has a high resolution high PPI display like the other two Mini Tablets that we test here. But shockingly, it still has the same small 63 percent Color Gamut as the original iPad mini and even older iPad 2."

So the best screen is the Amazon Kindle HDX, then the Google Nexus 7 (2013) and last the Apple iPad Mini.