Or, even worse, what happens when one quotes an noncontinuous part of the post? (uses ~snip~ or similar)
A person who wants to quote part of a post (or PM) could tell a tool which parts of the post (or PM) that he wants to quote, the tool could produce the exact text that should be copied (including its own implementation of "~snip~"), calculate the hash of the quoted message, and the source/reference of the message, and store this information in a forum database table (that is indexed by message ID). As long as the hash of the quote and message source match, the quote could be shown as verified.
So the original message would not need to be checked every time the new post is displayed. This would account for any possible issues regarding someone editing their post, breaking the quote verification. Although there could be a message displayed if a quoted message has changed since the verification request was generated.
This would basically mean implementing an (over)complicated editor that can handle the quotes like this. I hope that you don't realistically expect that users will use other (external?) tools for this purpose.
I don't have hopes for a such editor get implemented. I think that getting the new forum running/live even without such a feature is more important. That's why I've proposed only something simple.
Of course, if somebody has the time and willingness, sky is the limit, and then we can probably have an editor that handles the quotes, have notification if the quoted post was edited, have versions of the quoted post (for various uses, including the use for correctly showing/handling older quotes)... but, again, I don't realistically expect this come to live.