The reason why private-public key mechanism works is that it is really hard to get private key from public key. With quantum computers it is trivial task. In theory you could steal other people's bitcoins.

In reality I wouldn't be so enthusiastic about quantum computing exploding because of bitcoin. Private-public key encryption is used now everywhere and having quantum computer would give you benefit in lot of situations.

Prior to such an explosion, we'd have more interest in quantum-resistant public-key crypto: http://pqcrypto.org

One will likely end up being usable, and things will shift to it / others.