Bitcoin uses two cryptographic primitives: an elliptic-curve public-key signature for signing transactions, and SHA-256 for mining blocks and proof of work.
The public key signatures (ECSDA) could be broken by a quantum computer. Some level of protection is possible, because Bitcoin addresses are not "bare" public keys; they are hashes of public keys. But once a transaction is signed, the public key is visible and a sufficiently large quantum computer could recover the private key. This private key could be used to withdraw any remaining funds in the account. (Some Bitcoin experts recommend only using any given account once, for this reason and others.)
So, quantum computing would definitely endanger the ability for Bitcoin users to have control over their own accounts. (See the discussion here: https://en.bitcoin.it/wiki/Quantum_computing_and_Bitcoin)
The proof-of-work scheme is not so vulnerable. No quantum algorithm is known that breaks cryptographic hashes to the same degree. The best is Grover's algorithm which reduces a black box search, like a proof-of-work problem, from N queries to sqrt(N) queries. If only one person had a quantum computer, this would be a substantial advantage, and that person could probably use it for double-spending attacks. (Spend on block X, wait several blocks for confirmation, then use your advantage to go back to block X-1 and create a fork that is longer than the one where you spent your bitcoin. Then the new chain would be accepted by the other nodes.) However, if quantum computers are widely available, then the difficulty level of proof-of-work would increase, making this attack no longer feasible.