Qualsiasi and Qualunque have the same meaning and can be used in similar contexts. They are invariable. When they are placed before a noun, they mean "any" : Qualunque ragazza/ Qualsiasi ragazza When they are placed after a noun, they acquire a slightly pejorative meaning: Ho comprato una macchina qualsiasi / una macchina qualunque. ( I bought an unremarkable car, there is nothing special about it.) You can use them with plural nouns only if you place them after the noun: Non devi essere elegante per andare al museo, puoi indossare dei vestiti qualunque.

About the origin of qualsiasi: It comes from /quale che sia/ (Eng.: whatever it is), as you see it has the subjunctive (congiuntivo) mood in it. I can't guess about the origin of /qualunque/. I agree with Mona's explanation.

Hi Andy, don't worry, qualsiasi and qualunque are synonims, you can use either and they mean the same :) http://www.treccani.it/enciclopedia/qualsiasi-o-qualunque_(La_grammatica_italiana)/