Miners could mess with very recent state, but if you download enough blocks with full state to be confident of finality for the earliest one, and verify just those transitions, you're still safe.
If the incorrect blocks were published on the network, they'd be rejected by other miners, so you don't have to look far back unless (1) there's a sustained 51% attack on the whole network, or (2) you're a very juicy target and a huge miner can take over your internet connection, and you don't notice a large drop in difficulty.
If the incorrect blocks were published on the network, they'd be rejected by other miners, so you don't have to look far back unless (1) there's a sustained 51% attack on the whole network, or (2) you're a very juicy target and a huge miner can take over your internet connection, and you don't notice a large drop in difficulty.