  We construct the graded triple Lie commutator of cubic supermatrices, which
we call the quantum super Nambu bracket of cubic supermatrices, and prove that
it satisfies the graded Filippov-Jacobi identity of 3-Lie superalgebra. For
this purpose we use the basic notions of the calculus of 3-dimensional
matrices, define the Z_2-graded (or super) structure of a cubic matrix relative
to one of the directions of a cubic matrix and the super trace of a cubic
supermatrix. Making use of the super trace of a cubic supermatrix we introduce
the triple product of cubic supermatrices and find the identities for this
triple product, where one of them can be regarded as the analog of ternary
associativity. We also show that given a Lie algebra one can construct the
$n$-ary Lie bracket by means of an (n-2)-cochain of given Lie algebra and find
the conditions under which this n-ary bracket satisfies the Filippov-Jacobi
identity, thereby inducing the structure of n-Lie algebra. We extend this
approach to n-Lie superalgebras.
