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We introduce a series of $\Z_2^n$-graded quasialgebras $\bbP_n(m)$ which generalizes Clifford algebras, higher octonions, and higher Cayley algebras. The constructed series of algebras and their minor perturbations are applied to contribute explicit solutions to the Hurwitz problem on compositions of quadratic forms. In particular, we provide explicit expressions of the well-known Hurwitz-Radon square identities in a uniform way, recover the Yuzvinsky-Lam-Smith formulas, confirm the third family of admissible triples proposed by Yuzvinsky in 1984, improve the two infinite families of solutions obtained recently by Lenzhen, Morier-Genoud and Ovsienko, and construct several new infinite families of solutions.

ℤ₂ⁿ-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms