This Taiwanese company goes at the name of AU Optronics, and is one of the largest display panel makers, with a diverse flock of customers, which include Samsung itself. Despite that the CEO Paul Peng recently dismissed OLED as a viable alternative to LCD, citing yield and cost issues, AUO has been digging on the technology for quite a while now, and those efforts are definitely paying off.
AUO issued a press release that it has "successfully developed the world's highest-resolution 5.7-inch WQHD AMOLED panel." This Quad HD screens has 1440x2560 pixels of resolution, and breaks the 500ppi barrier, clocking in at exactly 513ppi. The highest-resolution panel on a phone so far is the 5.5" QHD 538ppi screen of the Oppo Find 7a, but this one is with LCD technology, and not even on sale yet. The other one announced with a QHD panel is the 6" Vivo Xplay 3S, which can be bought right now, but it is again LCD, and with 490ppi it doesn't reach the elusive 500 mark.
Thus, the 5.7" panel that AUO announced, is the highest-resolution AMOLED display unveiled to date, and the company said it has created a 6" Quad HD AMOLED panel, too. In addition, the Taiwanese have filled the niche for tiny AMOLED screens as well, announcing a 1.6" panel for wearable devices. Not only that, but AUO made a breakthrough with in-cell touch technology, releasing one of the thinnest AMOLED display packages, in the form of a 5" HD 720p screen with just 0.57mm thickness and 10-point touch, which is a prime candidate for ultrathin midrangers.
The moral of the story is that if AUO was able to muster a 5.7" Quad HD AMOLED display with 513ppi, then we can safely assume that the Galaxy Note 4 will sport at least such a screen, unless Samsung decides to go all fancy on us, and release it with a YOUM flexible panel.