transaction verification: Whole blockchain required?

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Do miners really require the whole blockchain to verify transactions? It would seem that it would be more efficient to have a database containing which address is allowed to spend how much, rather than traversing the whole blockchain to verify each transaction.
Related to this: If miners indeed use such a database, why not periodically "summarize" the blockchain? What I mean is: Old blocks are summarized using such a database. Then, the old blocks can be discarded and replaced with the database. It would seem that this scheme would dramatically decrease the size of the blockchain. Why is this scheme not implemented?