Good question and one that warrants a bit of research.
First off, what Qualtrics officially supports is the latest version of each major browser (Firefox, Chrome, Internet Explorer, Safari).
More information on that can be found here: http://www.qualtrics.com/university/researchsuite/research-resources/other-resources/faqs/#webbrowser
These are the browsers in which, theoretically, the QRTEngine should work as well, although whether they do so in all use-cases is difficult to predict. We have found that in Internet Explorer versions earlier than 9 the method becomes a bit less reliable (we have had a bug report probably concerning IE 8 at the time), and fully breaks with 7 and below. Other than that, it should work for all browsers that Qualtrics supports.
Please note that the method was not intended for use on mobile/tablet devices, and thus has not been tested on those devices.
Secondly, the engine can run in different modes. Native-highres, native, and timer. Native-highres is the preferred mode, as it both uses requestAnimationFrame (allowing for highly precise presentation durations) and a high resolution timestamp, which helps with several parts of the method.
If the high resolution timestamp is not available it falls back to using a much more crude timestamp, which will affect the precision.
If requestAnimationFrame is not available, it falls back to the ‘timer’ method, which is also less precise (we are in the process of putting a proper number on this), and generally for any experiment we’d recommend running the rAF version of the task, unless timing precision really isn’t a consideration.
Anyway, the highres timestamp is supported by the newest version of all major browsers. Chrome version 20+ supports it, Firefox version 14+, Safari only just started supporting it in version 8, Internet Explorer has supported it since 10.
requestAnimationFrame has been supported for a longer time, and currently is supported by all the newest versions of every major browser. Full support has been in Chrome since version 10, Firefox version 4, Safari version 6, and Internet explorer version 10 (please note that we disregarded partial support that happened while the specs were still settling down, the QRTEngine considers these partial implementations to be non-existant mostly).
These are the two main functions that we rely on while using the QRTEngine. There are differences between browsers that could cause certain functionality to break, but bugs due to these cross-browser differences should be encouraged to be reported, so that we can look at them and fix them. (Although we realize this is not always possible, it is sadly also unfeasible for us to test every version of every browser currently in existence).
In short, if timing precision is of low priority, use Qualtrics’ list of supported browsers (with the exception of Internet Explorer 8 and below). If timing precision IS important, use the lists from caniuse.com.
I hope this helps!