In cryptography there is quite a huge amount of literature on e-voting.
It allows quite a lot of complicated features: confidentiality of the votes, of course and anonymity of the voters but also the possibility for voters to check if their vote has been counted correctly when the result is published and much more. This question is a bit too broad to answer, here.
In practice, afaik none of these techniques are used by actual voting machines anyway. It is simply assumed that the machines are secure and do not cheat people. That's why there is a discussion about the system being tamper proof. In this context this is probably not about the entire voting process being tamper proof but about the actual physical machines.
This of course reduces e-voting to a bit of a credibility problem because nobody can figure out what these machines do, sometimes it is even illegal but everybody can count pieces of paper.
Note, cryptographic schemes have a similar credibility problem because non cryptographers cannot understand how they work.