AMA: Long-time DU lurker and democratic supporter, know everything there is to know about bitcoin

[Rassah]

88. It's not a big problem, apparently

Satoshi must have come from the future, or something, because when this issue was raised and examined, it turns out that he may have taken this into account, too. Basically, you have a private key (that lets you sign transactions), from that is derived the public key (which is the behind-the-scenes account number address), and from the public key is derived the bitcoin address that everyone sees. If all we used were public and public keys, then yes, using the public key a quantum computer can figure out the private key and steal the coins. But since only thing that's seen publicly is the bitcoin address, which is the hash of the public key (which is the hash of a private key), there is no way for a quantum computer to solve the private key. When you send money from a bitcoin address, then yes, your bitcoin address, public key, and private key signature are all publicly broadcast, and at that point a quantum computer can see the public key and get the private one, but as long as you don't send money, or if every time you send it, you send all remaining change to a completely new address, quantum computers won't be able to do anything.
Also, once we get close to quantum computers being a problem, the whole world will start switching to quantum-resistant algorithms.