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Apply a Hadamard gate to every qubit. The amplitude of $|0^n⟩$ becomesIt equals $1$ in case (I) and $0$ in case (II). A computational-basis measurement therefore returns $0^n$ with certainty in case (I) and never returns it in case (II). This is the Deutsch-Jozsa algorithm.

For the distributed problem, Alice preparesusing one query to her quantum oracle, and sends these $n$ qubits to Bob. Bob applies his phase oracle, producingThe function $h=f_A\oplus f_B$ is constant zero in case (1) and balanced in case (2). Bob applies $H^{\otimes n}$ and measures as above: outcome $0^n$ identifies case (1), while every other outcome identifies case (2), with certainty.