While quantum computers threaten some cryptographic schemes, it will just be the end of those schemes, not the end of cryptography in general (see: post-quantum cryptography). As long as we still have problems that are hard to compute and easy to verify, we can have cryptocurrencies. Quantum computers do not give "unlimited computing power" and do not threaten the existence of cryptocurrencies in general.
We know that Diffie Helmann, RSA and their elliptic counterparts are broken in quantum. That was essentially all we had. I know that 'lattice based cryptography' is a potential option, and believe there are more potential options. However, as far as I know it is not even clear whether these are safe against classical computers.
I should note that I don't know much more about 'lattice based crypto' than the name and the hope it is quantum resistent.
Memory-constrained algorithms, such as scrypt, should be quantum resistant based on what I know (which isn't much). If I'm not horribly misinformed, LTC mining should therefore be resistant. Again, LTC wallets would face problems.
[1]: https://eprint.iacr.org/2017/771.pdf [2]: https://crypto.stackexchange.com/questions/419/what-security... (NB: comments)
There is also no such thing as 'unlimited' computing power with regard to proof of work. Difficulty simply adjusts, and the cost of a network attack will always be related to the rewards of mining.
Remember the exitsanse of quantum computers does not prove P=NP, which is to say, even with quantum computers you can still have the class of problems which are difficult to solve but easy to verify which is the essential component of proof of work.
If it's a couple of thousand orders of magnitude, the first person with a working quantum chip might entirely dominate the hashpower, long before the chip is available commercially. So you'd end up concentrating hashpower into a couple of BigCorps's hands. Like, if I went back in time to the beginning of BTC with an ASIC I'd probably completely dominate the baby blockchain and outcompete the entire network.
With its use, the scarcity of numbers remain, that is, finding a clash that would allow to forge a transaction stays infeasible, and mining stays hard.
BTW while mining is so important with Bitcoin and Etherium, it's not a necessary part of a cryptocurrency. Mining is an incentive to keep doing blockchain validation. Some currencies exist without it (e.g. NXT).
Post-quantum cryptograph research is a very active area right now. If these researchers do not find a practical signature algorithm, PKI as we know it will change very significantly.
Ironically, blockchains might elevate in importance as a result, since the evolving blockchain is a useful construct for quantum resistance. It has cryptographic agility built in.
https://medium.com/@neha/cryptographic-vulnerabilities-in-io...
https://www.forbes.com/sites/amycastor/2017/09/07/mit-and-bu...
To make things even worse than rolling their own hash function, they're claiming the flaw was intentional! This makes them hostile and untrustworthy to the open source community if this is true and further erodes confidence they know what they're doing if this is their attempt at PR spin:
https://hackernoon.com/why-i-find-iota-deeply-alarming-934f1...
> Next, and in my mind most damningly, Sergey Ivancheglo, Iota’s cofounder, claims that the flaws in the Curl hash function were in fact deliberate; that they were inserted as ‘copy protection’, to prevent copycat projects, and to allow the Iota team to compromise those projects if they sprang up.
The creator requested an open, live debate to put these claims to rest. They haven't accepted and probably won't.