It's not a hard problem, it's an _impossible_ problem.
If there's any security hole between the BIOS, CPU, firmware, OS, app, a state-sponsored attacker will get it.
The US got into air-gapped systems. You think China (or Russia, Iran, ISIS, Europe, Anonymous, you name it) won't break in?
And what do you do if you discover that there was external vote manipulation? Hold re-elections?
And how easy is it for someone to delete, say, a couple thousand votes in a strategic state? It's hard to physically do it without evidence and a conspiracy, but if a hacker (or a simple bug) gets in, all bets are off.
That's why the paper-trail is so important