Our current player technology utilizes a steering script with it's own routines for creating controls, subtitles, and so on. It is designed in a way to extract the best of HTML5 or Flash video. Both HTML5 and Flash support 1080p.
For 4K quality or above, the host PC's ability to decode and display the video (at higher resolution) must be paramount as the GPU (Graphical Processor Unit) plays a vital role. Newer PC's and laptops are slowly getting there, but it would definitely choke older machines. Also, 4K will take up a huge amount of internet bandwidth.
NOTE: The same security controls apply to all video resolutions.