To prove that something happened after some time in a distributed way, just refer to something publicly know just before. Is that right? (Though I wonder how to make the time granularity smaller.)
What about to prove that something happened before some time in a distributed way ? How to make it more accurate, for example, an hour, a minute, a second, a milli-second?
Edit: To prove that you survives after a particular time, you just need to take a photo with a sheet of newspaper. It proves that you have lived some day after that newspaper was printed, though it cannot be certain which day.
It is quite distributed since you do not have to go for a notary office, etc., which is much more centralized, and is used widely therefore by those who have to refute rumors without having to go public.
It obsessed me how to do it conversely, that is, to prove that something exists or happens before a particular time. Go for a notary office is a centralized method to timestamp, which is able to prove something happens between two moments.
But to do it in a distributed way, it is best to do it yourself or with strangers anonymously. For example, buying a random newspaper is something that you can do with strangers anonymously.
Okay, let me explain the problem I am trying to solve, and hope you all may figure out if I went to the wrong direction. It is, though, something that is not quite relevant to the question itself, which is just a small, single link inside the chain, and that I didn't intend to talk about before. Here it goes.
I want to allow people to post messages publicly to a certain domain (a distributed storage system, details irrelevant) without further verification and any centralized service provider. (limited, once for all verification in Beta stage of the project is okay). But spamming will occur.
IP examination is not acceptable, since IP should not be collected, and not possible distributedly. I haven't found distributed CAPTCHA solution.
Proof of work won't work, since attackers can gather more CPU, GPU, and RAM, and create much more users and messages than normal people do, as do in bitcoin. Mutual random authentication between users will not work either.
I may limit user generation by introducing to the protocol invitation and collective punishment to the invitor and invitees, and give users quota per year.
So I have to prove that message is posted this very year, and people right now do not use quota from last year, one hundred years ago, or next year.