Can homeopathy cure cancer, Alzheimer's, MS or HIV?

Homeopathy has never been proven to cure anything, let alone very serious chronic illnesses like the ones you list. Homeopathy is the theory that if a substance usually provokes one reaction, then ingesting it in trace amounts will cause the opposite reaction. It has no scientific basis, only anecdotal evidence that is better explained by placebo effects.

In parts of Asia, and Middle East, homeopathy is a very renowned form of treatment. In fact in the olden days, in South Asia, for quite a long time homeopathic medicine WAS the only form of treatment.
My grandma's sciatica was treated ONLY by homeopathic medicine...even though her allopathic treatment had failed.
But I don't know if the list you have given can be treated. I don't think so though.

One should know its theory, which is rather in the form of philosophy. It really cures many chronic diseases, it takes time but the positive features are -being less costly and causing less or no side effects (if right medicines are chosen by a good physician).
If you go through the books, it doesn't have diagnosis, but the symptoms point to a medicine directly and very minor, seemingly unimportant points, will affect the medicine that is chosen. Hence, the question is wrong in the sense the disease enumerated are not treated, but the symptoms are treated, and patient usually gets some relief.
My suggestion is, when there is a definite & affordable medical or surgical treatment for a condition in allopathy, one should go for that.
About use of homoeopathy in south asia, I will like to point out that the origin of homoeopathy is in Germany and many european countries including England, recognize homoeopathy.