Only partially. You only need to know N blocks after a transaction to be N-certain a it is valid - that it's included in a block at all and propagated through the network implies the transaction is valid. If it weren't, other clients wouldn't accept the block.
You can never technically be completely certain a transaction isn't fake. You could be looking at a non-standard branch of the network, and you have no way of knowing it. So randomize your connections, wait for a few more blocks at reasonable difficulties, and you can be sure enough without having to work from the origin.
And (again, not 100% certain) I think you only need to store the hashes of each block to prove the chain is valid, and demonstrate a certain amount of work has been done (the difficulty of each). Each block solution builds only on the previous hash. That's just relatively tiny numbers, the entire list of block solutions would likely fit within 5MB with room to spare.
You can never technically be completely certain a transaction isn't fake. You could be looking at a non-standard branch of the network, and you have no way of knowing it. So randomize your connections, wait for a few more blocks at reasonable difficulties, and you can be sure enough without having to work from the origin.
And (again, not 100% certain) I think you only need to store the hashes of each block to prove the chain is valid, and demonstrate a certain amount of work has been done (the difficulty of each). Each block solution builds only on the previous hash. That's just relatively tiny numbers, the entire list of block solutions would likely fit within 5MB with room to spare.