I'm guessing that jackalopes have pronghorn antelope horns, because both are American. Pronghorn horns get called 'antlers' because they shed part of them every year, but they are not true antlers - scroll down to illustration of the skull.
That's pretty much where my knowledge of pronghorn horn/antlers ends. However, I can tell you why hardly any creatures have deer type antlers!
Deer antlers are one of evolution's freak accidents. To make antlers, deer have hijacked the wound-healing mechanism. The deer use the same kinds of biochemistry and physiology that you or I use to form a scab after cutting ourselves, or healing a broken bone. The trouble is, this is incredibly dangerous. Antler formation involves turning soft tissues into bone. If the process doesn't stop, first the antler, then the top of the deer's head, then the brain will be turned into bone. As you might imagine, having your brain turned to bone is ever so slightly fatal.
Deer have solved this in several ways:
- You shed the antler. Switch off the biochemistry that's calcifying everything to bone and switch on the regrowth biochem instead.
- You have structures on your skull called pedicles to keep those dangerous antlers the hell away from your tender little brain. Some types of prehistoric deer had ludicrously long pedicles, presumably because they hadn't quite finessed the timing of shedding. Sorry, can't find a decent image of those, but here is photoshopped impression of what Dicrocerus might have looked like. I don't think that's ancestral to modern deer, but it gives you an idea.
- At some point in their evolution, deer swapped their antler growth from being under control of the wound-healing hormones to being under control of the sex hormones. I'm guessing this is partly so rutting and maximum antler condition are synchronous. But it may also be for the deer to (metaphorically) say 'Done with sex now, so get rid of this life threatening thing now, now, now!'
So evolution of horns. Dead easy. Everything from dinosaurs to rhinos have done that. It's evolved multiple times, so it can be expected to evolve again in the future or on alien planets.
Evolution of an antler which you shed annually - much more complicated, and may be a 'once only' thing.