Haven't found much on the topic there, so asking away.

Let's say i have a practicable quantum computer or other device capable of rapid factorization of large enough integers.
What are the consequences to a developed bitcoin network?
Any way it could let me cheat in generation?
Any way it would let me cheat in transactions?

With a device of this kind i can get the private key from public key, right?
So after receiving a bitcoin from someone, can i subsequently successfully fake a transfer of all there was on his side?