Let's just host the downvote count somewhere else... all we need is a KV store, right? If we assume there are 10^10 videos there (it's surprisingly hard to find the exact numbers) and the downvote count won't go over 40 bits, then storing that data would need less than a terabyte in total.

(Only half-serious. I've seen a few "upvote this comment to dislike the video" and similarly clever workarounds already, so there does seem to be some bits of opposition and creative thinking left.)