An alternating Turing machine (ATM) is called a \(\Sigma_ m-TM\) if for all inputs the computation tree of M starts with an existential state and alternates along each computation path at most m-1 times. It is shown that a deterministic Turing machine with one work tape and with time bound \(O(n^ 3)\) can be simulated by a \(\Sigma_ 2\)-TM of the same type with time bound \(O(n^ 2 \log^ 2n)\). This implies \(\Sigma_ 2TIME_ 1(n)\not\subseteq DTIME(n^{3/2}/\log^ 6n)\) and NTIME(n)\(\not\subseteq DTIME_ 1(n^{1.22})\).