TITLE: Smallest witness for checking the primality of a number
QUESTION [7 upvotes]: In this link
https://primes.utm.edu/prove/prove2_3.html
it is stated that the smallest witness for a composite number is always 
less than $2ln(n)^2$ , assuming the extended Riemann-hypothesis.

Is there a known bound for a witness without making unproven asumptions ?

REPLY [2 votes]: For short, the answer is yes, but it is unbelievably weaker.
In particular, it changes the expected complexity from "polynomial" to "exponential".

Now, the long answer: by combining the Burgess' inequality with the Vinogradov's amplification trick (based on the Dickman function) it is possible to prove that $\eta_p$, the least quadratic non-residue $\!\!\pmod{p}$, is:
$$ \ll p^{\frac{1}{4\sqrt{e}}} $$
that is roughly $p^{\frac{1}{7}}$. Similar unconditional bounds are known for the least generator $\!\!\pmod{p}$. 
On the other hand, an unconditional result of Montgomery (that is an elegant improvement of a previous result of Chowla) shows that for an infinite number of primes, $$\eta_p \gg \log(p)\log\log\log(p)$$
holds.