In most states, no. While a handful of states have a simple, one- or two-page form that must be prepared, California is the only state that requires a separate application process rivaling the one required by the IRS. In California, federal tax-exemption does not eliminate state income tax liability until approval is received from the California Franchise Tax Board.
Organizations filing Form 1023-EZ are also required to file the entirety of the separate application to the California Franchise Tax Board.