Personally I think 4-wheeled bicycle-class vehicles should be allowed just like trikes are, especially since they can be safer in some situations than especially the most common type of trike, the delta, for a number of reasons covered in various other threads and websites.
But in my limited searching when I was originally planning a multi-wheel vehicle before I built CrazyBike2, I found that many states (Arizona included) only consider a bicycle-class vehicle to be such if it has "3 wheels or less in contact with the ground" (generally stated in some such fashion). Meaning, you could have four wheels on a bike, if two of them are like training wheels and are not touching the ground at the same time.
But if you have four or more wheels in contact with the ground (at the same time), it's no longer a bicycle...so then if you want it on the road legally, you have to build it to one of the other vehicle classifications, which usually forces you to step up to automotive levels of design, speed, braking, safety equipment, lighting, VIN, registration, insurance, emissions, crash testing (if applicable) etc. Lot of places seem to only allow lesser quads as off-road vehicles, and cannot be ridden/driven on public roads or ways.
Some places that do prohibit 4-wheel bikes probably don't enforce that, but it'd suck to spend all the money to buy or build one, ride it around, and suddenly have somebody with a bug up their bum ticket you for it, and no longer be able to use it.
Offroad it wouldn't matter...but on-road, youd' have to be sure they're allowed where you are or be prepared to deal with the consequences of enforcement, if any.