If I win the lottery, can I move to another state or Puerto Rico to save on taxes?
This would be no different than asking if you can live in one state and earn a paycheck, then move to another state with a lower tax rate before your tax bill is due so you can save on your taxes for that income.
Answer -- No
The tax on lottery winnings is based on the state where the lottery was held, because for legal purposes that's where the winnings are considered to have been earned for taxation purposes.
Also, changing where you live after earning money does not change your tax liability at all. You still owe the state where the money was earned the tax that is due.
I hope this helps.