  This paper begins with a review of the well-known KdV hierarchy, the $N$-th
Novikov equation, and its finite hierarchy in the classical commutative case.
This finite hierarchy consists of $N$ compatible integrable polynomial
dynamical systems in $\mathbb{C}^{2N}$. We discuss a non-commutative version of
the $N$-th Novikov hierarchy defined on the finitely generated free associative
algebra ${\mathfrak{B}}_N$ with $2N$ generators. Using the method of
quantisation ideals in ${\mathfrak{B}}_N$, for $N=1,2,3,4$, we obtain two-sided
homogeneous ideals ${\mathfrak{Q}}_N\subset{\mathfrak{B}}_N$ (quantisation
ideals) that are invariant with respect to the $N$-th Novikov equation and such
that the quotient algebra ${\mathfrak{C}}_N = {\mathfrak{B}}_N/
{\mathfrak{Q}}_N$ has a well-defined Poincare-Birkhoff-Witt basis. This allows
us to define the quantum $N$-th Novikov equation and its hierarchy on
${\mathfrak{C}}_N$. We derive $N$ commuting quantum first integrals
(Hamiltonians) and represent the equations of the hierarchy in the Heisenberg
form. Essential for our research is the concept of cyclic Frobenius algebras,
which we introduced in our recent paper. In terms of the quadratic form that
defines the structure of a cyclic Frobenius algebra, we explicitly express the
first integrals of the $N$-th Novikov hierarchy in the commutative, free, and
quantum cases.
