"We don't have any."
Translation:Non ne abbiamo alcuna.
It's a partitive prounoun, which means it takes the place of the thing of which a part is taken. This use of "ne" always concerns quantity; so, we can imagine someone asking "Avete mele?" at a frutteria. "Non ne abbiamo alcuna." "Ne" replaces mele and is translated "of them." Further, it is never used except when the quantity concerned is a part and not the whole. So, one wouldn't respond "Ne abbiamo tutte," but simply "Abbiamo tutte [le mele]."
"Non abbiamo alcuna" would literally translate (not mean, because the full meaning is hardly ever conveyed outside the context of the language) "we don't have any" which might get the message across, but is not elegant Italian. An Italian would know you're missing the partitive pronoun.
"Ne" has other very similar but less concrete uses:
As noted above, like with "qualcuno" for people, this is a singular usage. Translating "alcuno" as "any" just confuses English speakers into thinking the sentence should be structured differently. Tell yourself the "non ne abbiamo alcuno" means "we do not have a single one of those" and you'll understand that "a single one" and "alcuno" are the direct translation therefore it is singular.
Alcuno means "any" in the usual sense of an arbitrary subset. Qualsiasi is a much more specialized term. I think I've got it figured out, but feedback from a native speaker would be helpful.
Qualsiasi in front of a noun means "any" in the sense of "whatever it takes" or "whatever you've got." So "Mangio qualsiasi mela" means (I think) that whatever apple they give you, you'll rise to the occasion and eat it. (Perhaps this is a contest for eating rotten apples.) I don't think you can use this in a negative sentence.
After the noun, qualsiasi means ordinary. So "Mangio una mela qualsiasi" means that you aren't particular; you will eat any old apple.
One could respond to either sentence with "Non ne mangi alcuna." This would mean that the speaker thinks you aren't going to eat any of them.
I would say that "alcuno" is more like "some" to English speakers while "qualsiasi" is the true "any". So the sentence above, though it sounds awkward, would really be more like "do you have some?" "No, we don't have some," or even "no we don't have even a single one". Translating it as "any" just confuses the issue, even if the English sentence sounds better that way.
I don't know whether the object of "abbiamo" need be explicit, but my instinct would be to say an Italian would wonder what sort of part of the whole you don't have. For example, without "alcuna" I'm free to suppose you're only missing one of the apples; "alcuna" specifies just how much you lack, and I'm more likely to give you some of my apples, or at least feel sorry for you.
That's almost what I guessed too (I forgot the 'ne'), and Duo marked it wrong. Here's a discussion from Quora
Trying to piece it together, sounds like "Non ne abbiamo niente" would be "We don't have anything" whereas the "nessun" means "any" and describes the thing you're talking about in an implied sort of way, like "Do you have any apples?" "No, we don't have any"
"Avete delle mele?" "No, non ne abbiamo nessuna." o "Non ne abbiamo alcuna." I'm not an Italian though. Just my guesswork.