This is a note I made for the subject. I post it here for reference. If you find important posts that brought particular insight that is missing here or very well written posts, please point them out in this thread.
That one is able to find Proof of Stake blocks on multiple blockchains simultaneously causes the “Nothing-at-Stake” claim. Whether this is a problem has been discussed in the forums many times, notably What stops me from POS minting several different chains at once?, Punishing minting on multiple chains, and Pillow’s Peercoin Myths.
It’s been understood that anyone minting on multiple chains has slight advantage of getting POS reward earlier, therefore gaining compound reward. Whether this practice gives any advantage to mount an attack has been analyzed and questioned (here and here).
To eliminate the possibility of minting on multiple chains simultaneously Sigmike has suggested a solution in the cold minting thread and here, then explained here.
The solution, namely “Cancelling the best block when a duplicate stake is received” has been implemented, and was also announced in Sunny King’s weekly update.
There are other proposal of solutions, for example by