Not speaking specifically about quad propellers, but propellers in general though most of the "rules" I think would still apply.
It's been said propeller design is both science and black art. However, there are fundamental truths that are hard to get around. One of these is that the fewer the blades the more efficient the prop will be. This is due to interference drag - the following blade, if its placement is closer to the preceding blade, then it will also be closer to the disturbed air left by the preceding blade and this causes "interference drag". Single blade props are more efficient than two blade and two blades are more efficient than three.
So how come those WWII fighter planes had multi-blade props, some with 5 blades and even contra-rotating two props? They had to harness the increasing horsepower in a given prop arc. So max prop efficiency was not the major design goal.
All else being equal, and is seldom is, 3 blade props will run smoother than two. They also tend to have more static thrust and to climb slightly faster than two . . . but, they aren't faster. On a Mooney, a 3 blade will be 4 or 5 knots slower.
As blade length increases so does efficiency. But, few dispute that multi blade props look cool - - on airplanes it's called ramp appeal.
I'll be keeping two blades on my Phantom for now, flight time is more important to me than speed, and I suspect flight time will suffer with 3-blade props.
bumper