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Quantum complete intersections $A= A({\bf q, a})$ are Frobenius algebras, butin the most cases they can not become Hopf algebras. This paper aims to findbi-Frobenius algebra structures on $A$. A key step is the construction ofcomultiplication, such that $A$ becomes a bi-Frobenius algebra. By introducingcompatible permutation and permutation antipode, a necessary and sufficientcondition is found, such that $A$ admits a bi-Frobenius algebra structure withpermutation antipode; and if this is the case, then a concrete construction isexplicitly given. Using this, intrinsic conditions only involving the structurecoefficients $({\bf q, a})$ of $A$ are obtained, for $A$ admitting abi-Frobenius algebra structure with permutation antipode. When $A$ issymmetric, $A$ admits a bi-Frobenius algebra structure with permutationantipode if and only if there exists a compatible permutation $\pi$ with $A$such that $\pi^2 = {\rm Id}$.

Bi-Frobenius quantum complete intersections with permutation antipodes