The algorithm that compose the bitcoin's address are ECDSA and will be completely broken (you would be able to find one's private key with the public key). So you would be able to spend anyone's bitcoin.
The mining though is sha-256 based and is still "secure", in secure I mean it can't be simply reversed, but it still can be brute force. And since a quantum computer is exponentially more powerful people with QC would begin to mine like hell and the difficulty would rise to unseen levels. Since the difficulty is just merely an exponentionnal limitation the time to mine for an quantum computer will only grow linearly up until the maximum difficulty is reached (the maximum difficulty would require an hash of 0....all zeroes hash).
When this time come maybe it will block the chain (or maybe not) because a 0 hash maybe impossible to get, but in anyway massive damage would have been done to the blockchain.
This would happen if the quantum computer is introduced tomorrow, if we have a more progressive approach we can have time to change our algorithm to quantum ones, bitcoin can change his algorithm.