For starters, a system to be sure that a user or service is the same user or service it was previously.
That seems to be pretty trivial. What's wrong with a username/password combo (besides all the usual things) or, if you want to get a bit more sophisticated, with having the user generate a private key for himself?
You don't need a web of trust or any central authority to verify that the user named X is in possession of a private key which the user named X had before.
I'm more interested in if anyone's trying to solve it.
Well, again, the critical question is: What are you really trying to achieve?
If you want the online equivalent of the meatspace reputation, well, first meatspace reputation does not exist as one convenient number, and second it's still a two-argument function.
there's no attempts to run multi-dimensional reputation systems, to weigh votes by length of post or age of poster, spellcheck or capitalizations thresholds.
Once again, with feeling :-D -- to which purpose? Generally speaking, if you run a forum all you need is a way to filter out idiots and trolls. Your regular users will figure out reputation on their own and their conclusions will be all different. You can build an automated system to suit your fancy, but there's no guarantee (and, actually, a pretty solid bet) that it won't suit other people well.
I expect Twitter or FaceBook have something complex underneath the hood
Why would Twitter or FB bother assigning reputation to users? They want to filter out bad actors and maximize their eyeballs and their revenue which generally means keeping users sufficiently happy and well-measured.
What's wrong with a username/password combo (besides all the usual things) or, if you want to get a bit more sophisticated, with having the user generate a private key for himself?
In addition to the usual problems, which are pretty serious to start with, you're relying on the client. To borrow from information security, the client is in the hands of the enemy. Sockpuppet (sybil in trust networks) attacks, where entity pretends to be many different users (aka sockpuppets), and impersonation attacks, where a user pretends to be someone they are not, are...
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