Unless your state specifically exempts lottery winnings as income, you generally must pay tax on all worldwide income to your state of permanent residency. Therefore you must list your lottery winnings in your home state. If you have to pay tax in the source state (state in which you won the lottery winnings) then you may be able to get a credit for tax paid to another state on your home state's return.
Here is an example. I live in NJ. I took a trip to Las Vegas and won 10,000. I spent the whole 10,000 in Las Vegas on Hotel rooms, food, trips, etc. I still must list the 10,000 winnings on my NJ state return because I am a resident.
Please let me know if you need additional assistance.