= Solution
The <Affleck--Kennedy--Lieb--Tasaki state> places two virtual spin-one-half degrees of freedom at every site, puts neighboring virtual spins into <spin-one-half singlet state>[singlets], and projects the two virtual spins at each site onto their symmetric spin-one triplet. Its local tensor is equivalently proportional to $\sigma_x,\sigma_y,\sigma_z$ in the <spin-one Cartesian basis>, and its <Affleck--Kennedy--Lieb--Tasaki parent Hamiltonian> is
$$
H_{\rm AKLT}=\sum_jP^{(2)}_{j,j+1}.
$$
The one-dimensional <cluster state> is the simultaneous $+1$ eigenstate of the commuting stabilizers
$$
K_j=Z_{j-1}X_jZ_{j+1}.
$$
On a periodic chain it is obtained by applying a controlled-$Z$ gate on every neighboring pair of the product state $|+\rangle^{\otimes N}$.
After suitable blocking, the two states have the same nontrivial <projective virtual symmetry of a matrix product state> for the protecting group $\mathbb Z_2\times\mathbb Z_2$. The two virtual symmetry generators can be represented by anticommuting Pauli matrices, so they realize the nontrivial projective class in <group cohomology>. Consequently the AKLT and cluster states can be connected by a symmetry-preserving gapped path, or equivalently by a symmetry-preserving finite-depth local circuit: this is the <Symmetry-protected equivalence of the Affleck--Kennedy--Lieb--Tasaki state and cluster state>.
Both states therefore exhibit one-dimensional <symmetry-protected topological phase>[symmetry-protected topological order]. On an open chain their nontrivial virtual representation produces <protected edge state of a symmetry-protected topological phase>[protected edge degrees of freedom] and a characteristic degeneracy in the <entanglement spectrum of a matrix product state>. They do not have <intrinsic topological order>: if the protecting symmetry is discarded, either state can be connected to a product state by a finite-depth local circuit.
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