Doesn't quite work that way. The California Lottery's reporting requirements are the name of the winner and the store/location where the ticket was purchased. I've bought tickets while visiting other parts of California. If I could manage to win a big one (yeah - I know) while purchasing a ticket far from home, I would have the advantage that it would throw off people trying to find out where I live, and I would still have the advantage of no state taxes on the winnings. It says full name, but I haven't seen middle names announced, so that could just mean first and last name.
Can I be anonymous when I win the Lottery?
The California Lottery is subject to public disclosure laws that allow access to certain governmental records. Your full name, the name and location of the retailer who sold you the winning ticket, the date you won and the amount of your winnings, including your gross and net installment payments, are matters of public record and are subject to disclosure. The Lottery will not disclose any other personal or identifying information without your permission unless legally required to do so.
An out of state visitor winning in California would have an advantage in terms of potential harassment. But as I said earlier, it could create a problem for people with the same name. As for publicity, some people apparently are willing to talk, while others just go for the minimum reporting requirements. The California Lottery doesn't require that the winner pose for a photo as other state lotteries requires.