PoW has been shown, in bitcoin, to centralize, and we know the economic reason for that: "economies of scale".
No. PoW in bitcoin has shown to be a trustless, reliable proxy for elapsed time.
- cryptographically not very secure. Indeed, the cryptographic security resides solely in the need for an external attacker to do a *similar* amount of work than was needed to generate the security in the first place.
Firstly, 'cryptographic security' is the wrong term for what you are trying to describe. Secondly the security of a PoW chain is not based on doing a 'similar' amount of work, but to do more work than the rest of the miners in the network combined. That is indeed, 'vastly' more work.
But that is what a crypto currency should be: entirely determined by its owners. It is very strange to have a crypto currency that is depending on an external industry, and of which the users are not making up the consensus. A PoW coin is very much exposed to an external attack, while a PoS coin is cryptographically secure against an external attack. It can of course suffer *internal* attacks.
Again, you're misusing 'cryptographically secure' and even if we take your intended meaning, your statement is still wrong as PoS coins are vulnerable to a much broader range of attacks than PoW coins, both external and internal. Please see this thread for details:
https://bitcointalk.org/index.php?topic=1382241.0There is a paper describing a provably secure PoS chain, but even the author concedes that it can only be that way if a majority of honest nodes remain online. This is not a very resilient design, especially in the face of power cuts, wars and 'force majeure'.
Don't get me wrong, I'm not saying that bitcoin is a success - the network is congested beyond usability, but PoW remains the only trustless solution to the byzantine generals problem.
Cheers, Paul.