One problem: You fire up your blockchain software and it reaches out over the internet to find other servers and ask them if they have a copy of the blockchain.

But how does in know which copy is the real one and not one cooked up by someone like me showing I own half the coins?

With proof of work you can choose the one with the most work in it and that's hard to fake as the real bitcoin blockchain has billions of dollars worth of computer work in.

With proof of stake it's hard. If there is no real cost, fakers can make thousands or millions of fake chains and how do you know which is the 'real' one?

There are potential solutions but it's tricky. Like if you have www.stake-coin.com point to the real one, what if someone manages to take over the domain?