  This paper builds upon our recent work, published in Lett. Math. Phys., 112:
94, 2022, where we established that the integrable Volterra lattice on a free
associative algebra and the whole hierarchy of its symmetries admits a
quantisation dependent on a parameter $\omega$. We also uncovered an intriguing
aspect: all odd-degree symmetries of the hierarchy admits an alternative,
non-deformation quantisation, resulting in a non-commutative algebra for any
choice of the quantisation parameter $\omega$. In this study, we demonstrate
that each equation within the quantum Volterra hierarchy can be expressed in
the Heisenberg form. We provide explicit expressions for all quantum
Hamiltonians and establish their commutativity. In the classical limit, these
quantum Hamiltonians yield explicit expressions for the classical ones of the
commutative Volterra hierarchy. Furthermore, we present Heisenberg equations
and their Hamiltonians in the case of non-deformation quantisation. Finally, we
discuss commuting first integrals, central elements of the quantum algebra, and
the integrability problem for periodic reductions of the Volterra lattice in
the context of both quantisations.
