This is interesting, and it seems like a valid defence, though I can spot two weaknesses. First is that having more bitcoin-days is "easily" accomplished by having more than half the bitcoins. So you'd need 51% of hashing and 51% of bitcoins. Second is more subtle. If people get paid (e.g. civil servants) in the govt-approved blockchain, then they'll have to use that chain to spend their money. If there are lots of civil servants, then the govt blockchain will be correspondingly more important.
Yes, any proof-of-stake system of course helps a lot. In this context, the best thing about proof-of-stake, is that it forces the state/organization to provide owners with compensation for seizure of the blockchain (i.e. they have to buy out 51%). That more or less guarantees at least 51% of owners will be made better off by state/organizational control. By comparison, proof-of-work allows anyone to seize the blockchain without compensation.
The secondary advantage of proof-of-stake is that it makes seizure an order of magnitude more expensive. Consider the example I gave before. A lower bound on a proof-of-stake currency's market valuation is the discounted sum of txn fees associated with stake ownership.
Before we had:
It is worth pointing out that the investment is chump change and that the investment would be highly profitable. Say bitcoin replaces all card payments in the Euro area. That is about 54.8 billion Euros in 2008. Suppose that the average fee on bitcoin payments is 0.1%. Suppose that hardware depreciate at a 20% rate [slow for computer hardware], that labor costs are negligible [labor costs make monopoly easier], and that the interest rate is 5%. Assuming a competitive market, we have
capital rental costs=flow of bitcoin payments
(0.20+0.05)K=0.001*54.8 billion
K=$212 million.
Repeating this calculation using stake is difficult because the coin will also have uses as a medium of exchange and a store of value which add to its value. We can just ignore these and focus on fee collection.
capital rental costs=flow of bitcoin payments
(0.05)K=0.001*54.8 billion
K=$1.096 billion.
If we assume a lower interest rate (like in the economy today), then the difference is much more than five-fold. Suppose the real interest rate is 1%.
Then the attack costs are $260 million for PoW vs $5.480 billion for PoS, about a 20-fold increase.
It still won't stop a large state, but it would be a non-trivial obstacle for a private organization.