  There are two main approaches to the problem of realizing a $\Pi$-algebra (a
graded group $\Lambda$ equipped with an action of the primary homotopy
operations) as the homotopy groups of a space $X$. Both involve trying to
realize an algebraic free simplicial resolution $G_\bullet$ of $\Lambda$ by a
simplicial space $W_\bullet$ and proceed by induction on the simplicial
dimension. The first provides a sequence of Andr\'{e}-Quillen cohomology
classes in $H_{AQ}^{n+2}(\Lambda;\Omega^{n}\Lambda)$ for $n \geq 1$ as
obstructions to the existence of successive Postnikov sections for $W_\bullet$
by work of Dwyer, Kan and Stover. The second gives a sequence of geometrically
defined higher homotopy operations as the obstructions by earlier work of
Blanc; these were identified with the obstruction theory of Dwyer, Kan and
Smith in earlier work of the current authors. There are also (algebraic and
geometric) obstructions for distinguishing between different realizations of
$\Lambda$. In this paper we
  1) provide an explicit construction of the cocycles representing the
cohomology obstructions;
  2) provide a similar explicit construction of certain minimal values of the
higher homotopy operations (which reduce to "long Toda brackets"), and
  3) show that these two constructions correspond under an evident map.
