Arizona has some tough terrain, and water availability often limits wildlife populations. But what it may lack in quantity compared to other Western states, Arizona more than makes up for in quality.
Arizona has some of the best trophy mule deer, coues deer, elk, desert sheep, antelope, and bear hunting in the West. I apply for half-a-dozen species there every year, building points until I'm lucky enough to draw a tag.
To apply for licenses and be eligible to build bonus points, you have to buy a non-resident hunting license. This will set you back about $150. But if you're willing to make that commitment, you become eligible for bonus points across all species. ("Deer" is a single category for bonus points -- coues deer and mule deer aren't treated separately).
Draw odds for the best antelope and mule deer tags are staggering (1 in 100 or worse), there are some hunts that you can realistically draw, such as coues deer and elk. Sheep tags are, of course, not easy to come by, but it's conceivable that you could draw one.
I've only hunted Arizona once -- an unsuccessful spring bear hunt on the San Carlos Apache Reservation -- but an eager to return for another try.