Could eVoting on the bitcoin blockchain be done? I'm basically a crypto noob (and not an english native), so please be indulgent :)

I thought about if you could realize an eVoting system with bitcoin.

I see 2 major requirements: 


each citizen can only vote once
votes musn't be traceable


I imagined the government could give each citizen a private key (with their passport for example) and make the actual voting a multisig procedure, where you need to provide the government provided signature plus your own to sign your vote.

Now for my actual question (thx for your patience so far):

Is there some kind of math magic, that makes sure those 2 requirements are met? In other words: Could you, cryptographically use two private keys to sign a transaction, ensuring that one key was issued by a certain CA (= the government) without being able to trace this to the actual user (keeping the vote anonymous).
Yes, it could be done. However, it would work a little different from what you imagine.

The government can create a new blockchain operating under zerocoin rules. The government could then distribute one satoshi per citizen. This distribution could happen for example by letting each citizen visit a government office to present their passport and receive their satoshi. This satoshi on the particular blockchain constitutes permission to vote for this election.

After distributing the satoshis, the government would then announce the candidate parties in the form of zerocoin addresses. Each citizen could then vote for the party of their liking by transferring their satoshi to the account of the party they are voting for.

Finally, when the elections are over, the blockchain can be frozen at a particular block number and each party can reveal their received amount. These amounts can then be compared to elect a winner or counted to receive seats in a parliament.

This scheme satisfies both your stated properties of double-spending prevention and vote secrecy.

However, this is not a good voting scheme, as it has many problems. One important problem is coercion; a citizen can be forced to give away their vote at gunpoint.