The only thing that quantum computers can do is to speed up the calculation of SHA256 hashes. Even if its faster than normal computers by a factor of thousands, the ASICs would still be way faster than quantum computers. The difficulty will rise and the network would continue as per normal.

i dont think qunatum computers can speed up hashing, but anyway this is not what is meant by 'cracking' sha256.

Concerning quantum computers and cryptography, there are two totally different aspects.

1) quantum computers, if ever they come into existence with a lot of qubits (which I personally doubt, but ok), can

**TOTALLY CRACK the current public key systems** based on prime factorisation (RSA, Diffie-Hellmann) or based upon discrete logarithms in groups (elliptic curve crypto). The algorithm to do so is known, it is Shor's algorithm. By TOTALLY I mean totally: just ANY key can be cracked in a matter of milliseconds, on the condition that the quantum computer has more qubits than (a few times) the key length. If such a quantum computer exists, there is simply no difficulty in cracking the key, it doesn't take "days" or anything because the difficulty goes LOGARITHMIC with Shor's algorithm.

2) however, for hash functions, and symmetric crypto like AES-256, it can be shown that a quantum computer can AT BEST use Grover's algorithm to crack it. Grover's algorithm doesn't crack entirely a hash function, but essentially HALVES ITS BIT STRENGTH. So a SHA-256 hash (with 256 bits) would not require 2^256 trials like on a classical computer, but "only" 2^128 trials on a quantum computer, which is STILL IMPOSSIBLE to do practically. Most people think that quantum computers will, if ever they exist, run much slower than classical machines, so 2^128 trials on a quantum machine will be much harder to solve than 2^128 trials on a classical machine.

So while quantum computers can speed up hash function searching, they won't crack it entirely. The interesting thing is that under certain conditions, it has been established that Grover's algorithm is the best possible one on a quantum machine, to attack a random hash function.

==> big hash functions are still secure against quantum attacks ; most current public key crypto is totally broken by quantum attacks.

This is why it is somewhat strange, in the bitcoin protocol, to have hashed the public key to 160 bits, and not have kept the 256 bits. If the menace of a quantum attack were the reason for this, it would have been wiser to keep the 256 bit hash as an address instead of the 160 ripemd hash, because under grover's algorithm this would become only 80 bits secure, while the 256 bit hash would remain 128 bit secure under a quantum attack, which is the same level of *classical* security offered by the elliptic curve signature scheme - which wouldn't survive, by itself, a quantum attack. This is one of the peculiar crypto design "features" of bitcoin...