Understanding SPV client concept

I am reading the bitcoin original paper, the SPV section. It says “One strategy to protect against [fake blocks generated by attackers] would be to accept alerts from network nodes when they detect an invalid block, prompting the user’s software to download the full block and alerted transactions to confirm the inconsistency”. My question is how many blocks should SPV download to detect the inconsistency ? Just one ? Or should it download blocks preceding the alerted block, to verify them too?

Is there a plan for a Blockstack SPV client ?

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Perhaps @jude can help with this question.

I found there a really detailed description in the new docs:

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