What effects would a scalable Quantum Computer have on Bitcoin? A scalable quantum computer is a quantum computer that is easy to extend - adding more (q)bits of memory is not a fundamentally hard problem, and will happen. Or, alternatively, that it follows Moore's Law - its memory capacity and speed will increase exponentially over the years with technological advancement (the exponent might be relatively low).

Suppose such a Quantum Computer were constructed tomorrow - what would this mean for bitcoin?
You have a good discussion in:

https://bitcointalk.org/index.php?topic=133425.0

Basically, ECDSA is compromised, hashing isn't. With a quantum computer, you could easily deduce the private key corresponding to a public key. If you only have an address, which is a hashed public key, the private key is safe. Anyway, to spend a transaction, you need to send the public key. At that point you are vulnerable, but the attack is not straightforward. 

In general, quantum computers are not exponentially better than classical computers. You cannot access all the states in the superposition, only global properties. You can read http://www.cs.virginia.edu/~robins/The_Limits_of_Quantum_Computers.pdf to get a good idea of what they can and cannot do.