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A partial Hadamard matrix is a matrix $H\in M_{M\times N}(\mathbb T)$ whoserows are pairwise orthogonal. We associate to each such $H$ a certain quantumsemigroup $G$ of quantum partial permutations of $\{1,...,M\}$ and study thecorrespondence $H\to G$. We discuss as well the relation between the completionproblems for a given partial Hadamard matrix and completion problems for theassociated submagic matrix $P\in M_M(M_N(\mathbb C))$, in both casesintroducing certain criteria for the existence of the suitable completions.

The quantum algebra of partial Hadamard matrices