1) A QC is able to derive the private-keys for a wallet's public address, allowing for the theft of bitcoin
2) A QC able to perform the proof-of-work algorithm to mine new blocks at an order-of-magnitude faster rate than currently possible.
Fortunately for 1) (I think) it currently takes 2^512 (?) operations to break the private/public algorithm which is unfeasible to brute-force on normal hardware but a QC brings it down to 2^128 - but that's still on-the-order-of unfeasible - and in the event it ever does happen the blockchain could be changed overnight to use a new keying algorithm. And for 2) it would cause the blockchain difficulty to be pushed-up so high that people with QC machines would see the same ROI as today's industrial GPU and ASIC miners see - plus given that QC computers are horrendously expensive (think: billions of USD for a 50-bit general-purpose QC) it questions why you'd ever try to break Bitcoin as you'd already be a billionaire.