Everything has its own asymmetric key pair. For example, every voter, voting machine, poll worker, etc., should have at least one key-pair.
Whenever you do something like vote, you always get a receipt and immediately verify that the receipt is correct.
A correct receipt can be used to prove that the other party saw the content of the receipt and signed it with their private-key.
An incorrect receipt is useless. If you get one from a voting-machine, then you basically act the same way you would if the voting-machine gave you an out-of-order error message.
Use redundancy to protect against conspiratorial fraud. For example, the votes should all be electronically shared with the US government, the Republican party, the Democratic party, and whoever else – everyone can do their own count (super fast-and-easy, since it's electronic), and everyone should arrive at the exact same result without any errors. If anyone disagrees, everyone can show signed receipts to prove the truth; lies are easily and provably exposed.
Use chain-signed certificates to establish subordinate identities. For example, an official US voting-machine needs to prove that it's official, but it shouldn't have the US's main private-key (as that'd be a huge security liability). So instead, the voting-machine should have its own private-key, and then the US signs a receipt with the official private-key stating that the voting-machine's legitimate. Then the voting-machine can prove that it's legitimate by showing people the officially-signed receipt saying so.
Implementation complexity needs to be automated, open-source, and auditable.
Automation keeps all of this simple and easy-to-use. Sorta like how computers are complex, but most folks don't need to know how they work to watch Netflix.
Open-source and auditable so people can have trusted experts verify stuff for them. For example, US Republicans would probably feel more secure if the Republican party independently verified that their voting credentials were good, and people who don't trust a single party could ask multiple parties to all verify correctness.
Initial adoption and social ails are the complicated hurdle. It's easy to under-/over-estimate how bad they might be, or misjudge how they might play out, so I'm hesitant to assume too much.
Terminology: "Web of trust".
Most of what was sketched above is more web-of-trust than block-chain. Block-chain elements could be added in if appropriate, but for reasons that seem silly to worry about discussing here, I doubt that'd make sense in this application.