In the United States, at least, the standard noun/adjective correspondence is nation/national ~ state/state, as Edwin Ashworth suggests in a comment above.
That correspondence may not look terribly appealing, but you can demonstrate to yourself that it is accurate as applied to most situations by considering such pairs as "national elections/state elections," "a national bank/a state bank," and "a national problem/a state problem."
In the OP's example, saying that a building "was deemed to have state significance" is exactly how that idea would be expressed (in passive voice) in a typical U.S. newspaper's account of the decision. In my view, none of the other options—statal, statial, or statewide—suggested above suits the context of the situation particularly well. You could argue, however, that statewide is more suitable if the main point you're trying to make is that the building has significance to people across the state, whereas state is more suitable if your main point is that the building has significance to the state taken as a collective entity.