Is there a relationship between Entropy of a fininte distrete probability distribution and the squre sum of the values of probability mass function of that distribution? Sorry for the long title. What I mean is that for two vectors (a_1,...,a_n) and (b_1,...,b_n) with the property $a_i,b_i \geq 0 $ and $ \sum a_i =\sum b_i =1$. 
If $ -\sum a_ilog(a_i) > -\sum b_ilog(b_i) $ implies  $\sum a_i^2 < \sum b_i^2$ or something similar?
 A: Of course, in general it is not true that inequality between entropies implies the same inequality between $\ell^2$ norms. Although this is true for $n=2$, it fails already for $n=3$ (as the entropy $H(p_1,p_2,p_3)$ is obviously not constant on the level curve determined by conditions $\sum p_i=1$ and $\sum p_i^2=Const$). 
Nonetheless, there is a deep link between these two quantities. In order to explain it, it is better  to somewhat change the viewpoint. Namely, given two probability distributions $P=(p_1,\dots,p_n)$ and $Q=(q_1,\dots,q_n)$ (for simplicity I assume that all $p_i,q_i$ are strictly positive), the corresponding Kullback-Leibler deviation is defined as 
$$
I(Q|P) = \sum p_i \log \frac{p_i}{q_i} \;,
$$
and the so-called information energy as
$$
\chi^2(Q,P) = \sum \biggl(\frac{p_i}{q_i}-1\biggr)^2 q_i \;.
$$
Both these quantities measure "closedness" of $P$ and $Q$ (although they are not distances) and are monotone invariants of the pair $(Q,P)$ (they do not increase under quotient maps). If $Q=U_n$ is the uniform distribution, then these quantities are, up to linear rescaling, precisely the entropy and the $\ell^2$-norm of $P$, respectively, as
$$
I(U_n|P) = \log n - H(P)
$$
and
$$
\chi^2(U_n,P) = n \sum p_i^2 - 1 \;.
$$
Now, the link between $I$ and $\chi^2$ is provided by the fact that in naturally defined infinitesimal limits the Kullback-Leibler deviation $I$ and the information energy $\chi^2$ coincide
and produce the Fisher information metric. One can read more about this in the corresponding wiki articles, in the old book Statistical decision rules and optimal inference by Čencov (AMS, 1982), or in more recent publications on information geometry.
