Unless they are the fancy unbleached 100% cotton organic tampons, I doubt they would compost. Tampons and other menstrual products aren't required to have an ingredients list, so who knows what's in there. I believe most tampons have synthetic as well as natural fibers. we

Personally, my menstrual cycle has been 100% reusable since 2003, and I much prefer reusables to disposables on a comfort level. I mostly use cloth pads, but I use a cup sometimes also. I used to be the opposite - mostly cup, sometimes cloth pads.