MN has paper ballots and also has automatic recounts of random precincts within each county regardless of the vote margin. We have had two major state wide recounts in recent years and both have been successful - they have been transparent and fully auditable.

http://www.ceimn.org/ceimn-state-recount-laws-searchable-database/states/Minnesota

There are other issues about whether certain people are eligible to vote, and how to handle that on election day and what to do if it is later determined that someone who shouldn't have voted did vote. (Answer: not much you can do, since their vote is anonymous... BUT there are VERY VERY few cases where this happens and is discovered.)