This paper studies the problem of minimizing a homogeneous polynomial (form) f(x) over the unit sphere Sn-1={x∈?n:‖x‖2=|1}. The problem is NP-hard when f(x) has degree 3 or higher. Denote by f_min (resp. f_max) the minimum (resp. maximum) value of f(x) on Sn-1. First, when f(x) is an even form of degree 2d, we study the standard sum of squares (SOS) relaxation for finding a lower bound of the minimum f_min:max? γ s.t. f(x)-γ·‖x‖22d? is SOS.Let f_sos be the above optimal value. Then we show that for all n≥2d,1≤fmax?-fsosfmax?-fmin?≤C(d)(n2d).Here, the constant C(d) is independent of n. Second, when f(x) is a multi-form and Sn-1 becomes a multi-unit sphere, we generalize the above SOS relaxation and prove a similar bound. Third, when f(x) is sparse, we prove an improved bound depending on its sparsity pattern; when f(x) is odd, we formulate the problem equivalently as minimizing a certain even form, and prove a similar bound. Last, for minimizing f(x) over a hypersurface H(g)={x∈?n:g(x)=1} defined by a positive definite form g(x), we generalize the above SOS relaxation and prove a similar bound.