  We analyze rates of approximation by quantized, tensor-structured
representations of functions with isolated point singularities in ${\mathbb
R}^3$. We consider functions in countably normed Sobolev spaces with radial
weights and analytic- or Gevrey-type control of weighted semi-norms. Several
classes of boundary value and eigenvalue problems from science and engineering
are discussed whose solutions belong to the countably normed spaces.
  It is shown that quantized, tensor-structured approximations of functions in
these classes exhibit tensor ranks bounded polylogarithmically with respect to
the accuracy $\epsilon\in(0,1)$ in the Sobolev space $H^1$. We prove
exponential convergence rates of three specific types of quantized tensor
decompositions: quantized tensor train (QTT), transposed QTT and Tucker-QTT. In
addition, the bounds for the patchwise decompositions are uniform with respect
to the position of the point singularity. An auxiliary result of independent
interest is the proof of exponential convergence of $hp$-finite element
approximations for Gevrey-regular functions with point singularities in the
unit cube $Q=(0,1)^3$. Numerical examples of function approximations and of
Schr\"odinger-type eigenvalue problems illustrate the theoretical results.
