Nondeterministic speed-up of deterministic computation
You should not expect an exciting speed-up. We have

$$\mathrm{DTIME}(f(n))\subseteq\mathrm{NTIME}(f(n))\subseteq\mathrm{ATIME}(f(n))\subseteq\mathrm{DSPACE}(f(n)),$$

and the best known simulation of deterministic time by space is still the Hopcroft–Paul–Valiant theorem

$$\mathrm{DTIME}(f(n))\subseteq\mathrm{DSPACE}(f(n)/\log f(n)).$$

Thus, nondeterminism or alternation is not known to give speed-up by more than a logarithmic factor. (I suspect no super-linear speed-up is known either, though I’m not sure if the HPV theorem can’t be made to work with ATIME in place of DSPACE.)