What I'm going to say won't change the fact that this story is likely false, but there technically is a way that you could use the "blockchain" to secure these ballots with a form of a "watermark".

The watermark obviously couldn't make up the blockchain, because as you said, a blockchain is a regularly updated chain of digital blocks. However, each ballot's watermark could theoretically store a cryptographic key, and then said cryptographic key could have been added to a blockchain stored on a government system somewhere, as the ballot is mailed out. None of this GPS tracking nonsense, none of this quantum nonsense. What this would get you is a way to verify whether a ballot is real, and if it was sent to the right person, by checking the cryptographic key in the ballot as it's scanned, and then compare it to the cryptographic key in the blockchain on the government secure server.