# are multi-qubit controlled Z gates (CZ,CCZ,CCZ,...) symmetric

The CZ gate is known to be symmetric : $$CZ(a;b)=CZ(b;a)$$;

what about $$CCZ(a,b;t) \stackrel{?}{=} CCZ(a,t;b) \stackrel{?}{=} CCZ(t,a;b) \cdots$$

same question for $$CCCZ$$ gates..

I think the answer is yes, but I'd like to see a reference or better yet a nice way to prove it.

A $$\mathsf C Z$$ gate rotates the phase of $$|11\rangle$$, and does nothing to the three other basis states.
A $$\mathsf C^{n-1}Z$$ gate will rotate the phase of the $$|11\ldots 1\rangle$$ state, and will do nothing to the $$2^n-1$$ other states.
If any of the $$n$$ qubits are $$|0\rangle$$, no phasing occurs.
Therefore such gates are symmetric and invariant under permutation of the qubit indices. For $$n=2$$, we often emphasize this by circuit diagrams having the $$\mathsf C Z$$ gate be symmetric.