To clear up some confusion: You have to cash your Powerball ticket in the state where you bought it and you will be paid by that state. This applies even if your own state also participates in the Powerball lottery.
The winnings will be treated as income sourced within the state where you won it and will be taxed according to the rules of the state where you won it and the state where you are a resident.
If you live in Texas (which has no income tax) and win the lottery in Illinois, you will pay no tax to Texas (since it has no tax) and pay the regular non-resident income tax in Illinois (which currently is 3.75% after deducting a small exemption). The other way around, if you are an Illinois resident and win a lottery in Texas, you will pay the full Illinois state income tax on the winnings and nothing in Texas.
California is a more interesting case. California exempts California lottery winnings (but not out-of-state lottery winnings) from state income taxes. So if a Texas resident won the lottery in California, they would not pay any income taxes to any state. But if a California resident won the lottery in Texas, they would pay California taxes on their winnings at the regular California resident rate.
A California resident who won the Illinois lottery would pay the regular income tax rate to both Illinois and California, but they would receive a credit on their California income tax return for the taxes paid to Illinois (not to exceed the amount of tax that California charges on the lottery money).
California and Arizona reverse the regular rules about claiming a credit for out-of-state taxes. If a California resident won the Arizona lottery, Arizona would withhold 6% of the winnings. The California resident would then have to file an Arizona non-resident tax return and a California resident tax return and claim the Arizona winnings as income in both states. But Arizona (not California) would grant a credit for taxes paid to California. Since California state income taxes are generally higher than Arizona taxes, this would usually result in getting a refund of all of the taxes withheld by Arizona and paying the full California rate.