We define quantum automorphisms and isomorphisms of Hadamard matrices. We
show that every Hadamard matrix of size $N\ge 4$ has quantum symmetries and
that all Hadamard matrices of a fixed size are mutually quantum isomorphic.
These results pass also to the corresponding Hadamard graphs. We also define
quantum Hadamard matrices acting on quantum spaces and bring an example thereof
over matrix algebras.

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