The notion of spectrum for first-order properties introduced by J. Spencer
for Erdos-Renyi random graph is considered in relation to random uniform
hypergraphs. In this work we study the set of limit points of the spectrum for
first-order formulae with bounded quantifier depth and obtain bounds for its
maximum value. Moreover, we prove zero-one k-laws for the random uniform
hypergraph and improve the bounds for the maximum value of the spectrum for
first-order formulae with bounded quantifier depth. We obtain that the maximum
value of the spectrum belongs to some two-element set