You can't keep a hedgehog as a pet in Arizona. You can keep a hedgehog in Arizona if you get a Wildlife Holding license from the Dept. of Fish and Game, but you can only get a license if you want a hedgehog for scientific, educational, or humanitarian purposes. The official language:
A wildlife holding license shall authorize an individual to possess, transport, import, display for educational purposes, photograph for commercial purposes, purchase, propagate, export, give away, or euthanize either restricted live wildlife or live wildlife lawfully held under a hunting or fishing license for any of the following purposes: advancement of science, wildlife management, or promotion of public health or welfare; Education: photograph for a commercial purpose live wildlife that is lawfully possessed; to give humane treatment to restricted live wildlife that has been abandoned or permanently disabled; or to lawfully possess restricted live wildlife that was possessed under another special license, and the primary purpose for that special license no longer exists.
Hedgehogs are "restricted wildlife" in Arizona. The main concern is that freed hedgehogs would find the climate very amenable, and since they have no natural predators here, a hedgehog population could disrupt the local (fragile) ecosystem. As I explained to DD, "It would be bunnies in Australia all over again."
DD was disappointed even though we hadn't even discussed the possibility of getting a hedgehog. The kind officer at Fish & Game told me there are some hedgehogs at the Phoenix Zoo, but he may have meant the World Wildlife Zoo. I'll have to look into that so she can get a hedgehog fix from time to time.