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In brassard et al. Amplitude Amplification work, they define the Q operator as

$$\mathbf{Q} = -AS_{o}A^{-1}S_{\chi}$$

where $S_{o}$ is an operator which flips the sign of the $|0 \rangle$ state.

Which is basically a diagonal unitary matrix (in the computational basis) with -1 on the first diagonal element.

I was wondering, isn't Quantum amplification's quantum speedup hindered by the realization of $S_o$ when the number of qubits in the circuit is too big? Based on Barenco's et al work (Elementary gates for quantum computation), isn´t the number of gates required for a $n$-qubit controlled gate exponential in $n$?

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No, not at all. Barenco et al.'s work is primarily saying that for certain specific gates (the multi-controlled phase gate, as required for $S_0$, being one), you can construct them in a time that is polynomial in $n$. Yes, the general case might require exponentially many, but not for every case, and it's the small number of non-exponential cases that we do our best to use.

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