Bare with me with this reply has I've been working on it for a while and have dropped it and came back to it a few times over the last few days. Its also taken me a while to get this all down. This discussion actually prompted me to log in to the forum after a while because this quality discussion is a rarity these days on this forum.

What I want to address is the different types of solutions which are currently either being developed or are fully developed and been deployed elsewhere. First I'll talk about the quantum resistant ledger and why Bitcoin doesn't need this and in fact I prefer the way Bitcoin is dealing with the whole quantum computer threat. Lets be clear and say quantum computers actually already exist and are already being used for multiple different things other than cracking algorithms and encryption. However just like its been discussed in this thread quantum computers are a number of years from becoming good enough to be able to threaten most encryption and certainly Bitcoins method of encryption. Estimations by large companies within the field has predicted we'll be seeing quantum computers with 2500+ qubits by the year 2025 but like many of you said these will be only just developed and will require a lot of testing and certainly won't be available to the average joe. However just because the mass population doesn't have access to these superior quantum computers doesn't mean its still not a threat. Governments which are probably funding these quantum computers are known to be pretty hostile towards Bitcoin and could use this to their advantage along with other things on their agenda. What some altcoins have done is included a quantum resistant ledger right from the get go in order to try and appeal to those that are misinformed. Implementing a quantum resistant ledger is all good when the altcoin its protecting is only worth a few cents but imagine implementing a untested and most importantly unproved quantum resistant ledger in a multi million pound industry like Bitcoin. This is the reason why Bitcoin developers and us as users of the software should be encouraging the development of Bitcoin to thoroughly test anything before its added to the "mainframe". The quantum resistant ledger I'm talking about has been implemented by a number of different altcoins but we are still unsure whether they will be ready and scalable without causing too much disruption to the value and of its users. This is why Bitcoin is one of the more respected cryptocurrencies out there because everything which is implemented is thoroughly tested and isn't just trying to appeal to people with new sparkly features. The quantum resistant ledger for example is a complete waste of time right now and isn't much more than a gimmick. It doesn't provide any more level of security than Bitcoin does and when quantum computers are able to break the algorithm Bitcoin uses it will then switch to a quantum resistant one which could in fact be better than the current quantum resistant ledgers we are seeing because its been tested over a number of years instead of just developed and thrown in there even when its not needed.

Despite these quantum resistant ledgers being gimmicks currently because they aren't providing any more security than traditional cryptocurrencies its at least a good idea to provide proof of concepts to the developers of Bitcoin and they can improve on the existing quantum resistant algorithms.The current quantum resistant solutions out there are mostly using eXtended Merkle Signature Scheme a hash-based digital signature system which allows reusable addresses and this is where I think Bitcoin could implement a less invasive algorithm onto the network. The problem with reusing addresses is once they have broadcast themselves onto the network they are then vulnerable to an attack from a quantum computer because they have exposed their public keys onto the network. This hash would then be suspect to quantum computers by using factoring to break the encryption and this is where I think the network could be improved without implementing a fully quantum resistant ledger by only allowing the use of an address once. So you could receive x amount on one address and then the wallet software automatically assigns that to a different address without broadcasting it to the network. I think this is possible and should be the only time an amount isn't broadcast to the network or only allow addresses to be used once by allowing them to receive coin and send from it once this would reduce the probability of a quantum computer finding the public key and attempting to crack it. We could actually do this in the current implementation of Bitcoin but not many people do and they simply reuse addresses even when its recommended to only use addresses once to avoid privacy issues. However if this was fully implemented into the network as a standard I think that would solve most of the problems. Its not completely safe but doesn't have a massive impact like implementing an entire new algorithm. At least for now we all know that Bitcoin will eventually have to adopt a new algorithm to keep up with the hardware being developed but so will many other things in the world.

Secondly lets talk about factoring and how quantum computers actually do it because I've seen this mentioned in a few of the replies but none of them have really gone into enough depth to justify mentioning it. So quantum computers are exceptionally fast at a few things and one of them is factoring. Factoring is used to crack conventional cryptography and this will be the route that quantum computers will take if they were to ever break the algorithm of Bitcoin but just mentioning factoring isn't really explaining how quantum computers are exceptionally good at it. Well quantum computers are very good at solving Discrete Fourier transform which in mathematics is converting a finite sequence of equally spaced samples of a function into a same the exact same length sequence of equally spaced samples of the discrete time Fourier transform which is a complex valued frequency. Discrete time Fourier transform is used to analyze samples of a continuous function. Discrete time is called that because it handles discrete data which their intervals are units of time basically. So using these functions quantum computers factor against the algorithm to find the solution. We have probably studied factoring at one point in our lives which includes multiplication however the factoring that quantum computers are doing is on a completely new level to that. Here's an example of a factoring problem:

The folllowing factoring problem

would be solved by comparing common factors and using multiplication groups There's a great explanation already out there which outlines this problem and provides the sequence of the process to determine the solution: https://quantumcomputing.stackexchange.com/questions/1383/what-makes-quantum-computers-so-good-at-computing-prime-factors

By increasing the amount of qubits a quantum computer has we are effectively making it quicker at solving these problems by using the above mentioned method. This process is a lengthy one using current modern day computers but the idea behind quantum computers is once they have hit 3000 qubits they will be able to break most current day algorithms within a matter of seconds.