Abstract: Linear Programming (LP) and Semidefinite Programming (SDP) have been used extensively to obtain approximation algorithms for combinatorial optimization problems. One possible avenue of improvement for such algorithms is the use of "lift-and-project" techniques, which, given an Integer Program (IP) over $n$ 0-1 variables, define a hierarchy of LP or SDP relaxations which converge to the convex hull of the original IP solutions. Since the LP and SDP relaxations arising from the $k$th level of such hierarchies can be solved in time $n^{O(k)}$, one would hope to obtain improved approximations after a constant number of iterations. Unfortunately, for several problems (most prominently Vertex Cover), it has been shown that for various LP and SDP hierarchies, the quality of approximation (the integrality gap) does not improve substantially even after many rounds of lift-and-project. Here we consider SDP relaxations arising from the third level of the Lasserre hierarchy, and show that they yield improved approximations for two combinatorial problems. The first problem is Maximum Independent Set in 3-uniform hypergraphs. The previous best approximation for this problem was given by Krivelevich, Nathaniel and Sudakov, who gave an algorithm finding an independent set of size $\tilde{\Omega}(n^{6\gamma-3})$ for 3-uniform hypergraphs with an independent set of size $\gamma n$ (but no guarantee when $\gamma \leq 1/2$). We show that for some fixed constant $\epsilon > 0$, given a hypergraph containing an independent set of size $(1/2-\epsilon)n$, we can find an independent set of size $\Omega(n^{\epsilon})$. The second problem we consider is that of legally coloring 3-colorable graphs using a minimum number of colors. We give an algorithm which finds an $O(n^{0.2072})$-coloring given a $3$-colorable graph, improving upon the work of Arora, Chlamtac and Charikar. ----------------------------------------------------------------------------