Computational complexity of computing homotopy groups of spheres

Francis Sergeraert and his coworkers have implemented his effective algebraic topology theory in a program named Kenzo. It seems capable of computing any $\pi_n(S^k)$ (in fact homotopy groups of any simply connected finite CW complex), although I don't know how far it is feasible. For instance $\pi_6 S^3$ is computed in about 30 seconds. In a 2002 paper, they mention other algorithms by Rolf Schön and by Justin Smith, not implemented at that time.

It is shown by D. J. Anick in The computation of rational homotopy groups is #℘-hard. Computers in geometry and topology, Proc. Conf., Chicago/Ill. 1986, Lect. Notes Pure Appl. Math. 114, 1–56, 1989. that, well, the computation of rational homotopy groups is #p-hard.

Here's a very useless algorithm due to Kan: Let $G(S^n)$ be the simplicial group that is Kan loop group of the $n$-sphere. In each simplicial degree, it is a free group. (This simplicial group has the homotopy type of the based loop space of $S^n$, so its homotopy groups compute the homotopy groups of $S^n$ shifted by one degree.)

For each simplicial degree $k$ define $N_k(S^n) \subset G_k(S^n)$ to be the intersection of the kernels of all but the last face maps $d_i: G_k(S^n) \to G_{k-1}(S^n)$. Then the last face map is a homomorphism $d_k: N_k(S^n) \to N_{k-1}(S^n)$. Moreover, the simplicial identities show $d_kd_{k+1}$ has constant value $1$, so we get a non-abelian chain complex of free groups. Its "homology," by a result of Kan, computes $\pi_*(S^n)$.

To get an algorithm for computing this homology, recall that the proof of the Nielsen-Schrier theorem gives a system of generators for the subgroup of any free group. So we obtain a system of generators for $N_k(S^n)$ as well as a system of generators for the image of $d_{k+1}$. So in principle we obtain a method for computing the homotopy groups of spheres.

In Kan's paper, $\pi_3(S^2)$ is computed in this way, and it takes several pages––so it's not a very good algorithm!