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"Succinct circuit representation" on Turing machines?

A circuit is not represented "on" a Turing machine, rather a representation of the circuit is provided as input. It is straightforward to represent a circuit as a bit-string. For instance, ...
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Representing classical circuits with quantum gates

No, it's obviously not equivalent, because computation on a quantum computer is not known to be equivalent to computation on a classical computer. Also, it's not clear what you mean by a "quantum ...
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Acyclic boolean circuit (DAG)

If your circuit can contain registers (or some other form of memory or state) (e.g., sequential logic), then yes, you can. You store the value of all variables in the registers, and you build a ...
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