I'll trust electronic voting machines once they start printing out receipts detailing how someone voted. Hell, you can even take all the responses by said voter, assign a value to them, then hash the entire thing into a simple string of numbers as a receipt.

Voters (SSN+Value1+Value2+Value3+etc) -> Sha-512 hash and done.

To recreate the exact hash you would need the exact values the voter chose + their SSN. A single digit off will alter the entire output hash.

It's really that simple.

Without a hardcopy, it's just too simple to tweak a database one way or another with a script or a keystroke or two.