BenQ W1070. 1080p.
One of the main reasons I got the W1070 was to use it with PS4. I tried Infamous Second Son on my previous 720p projector, and it just lacked that oomph, if you know what I'm sayin... Part of it, was the fact that my previous projector had a bulb with a ton of hours on it, and it was quite dim at this point. Still, going to the W1070 was like a night and day difference. The vibrancy while playing Second Son is off the charts.
I most recently was playing the Destiny Alpha on the W1070 and that looked great as well. I've also tried the Trine 2 demo in 3D on it, and it was pretty good, but the 3D Blu Ray movies are much more impressive. Avatar looks almost as good as going to the local cineplex.
As for Sync issues, sometimes when the PS4 changes resolutions, the projector will loose the signal for a second or two, and then everything will be fine.