  The framework of algebraically natural proofs was independently introduced in
the works of Forbes, Shpilka and Volk (2018), and Grochow, Kumar, Saks and
Saraf (2017), to study the efficacy of commonly used techniques for proving
lower bounds in algebraic complexity. We use the known connections between
algebraic hardness and pseudorandomness to shed some more light on the question
relating to this framework, as follows. 1. The subclass of $\mathsf{VP}$ that
contains polynomial families with bounded coefficients, has efficient
equations. Over finite fields, this result holds without any restriction on
coefficients. Further, both these results also extend to the class
$\textsf{VNP}$ as is. 2. Over fields of characteristic zero, $\mathsf{VNP}$
does not have any efficient equations, if the Permanent is exponentially hard
for algebraic circuits. This gives the only known barrier to ``natural'' lower
bound techniques (that follows from believable hardness assumptions), and also
shows that the restriction on coefficients in the first category of results
about $\mathsf{VNP}$ is necessary. The first set of results follows essentially
by algebraizing the well-known method of generating hardness from non-trivial
hitting sets (e.g. Heintz and Schnorr 1980). The conditional hardness of
equations for $\mathsf{VNP}$ uses the fact that pseudorandomness against a
class can be extracted from a polynomial that is (sufficiently) hard for that
class (Kabanets and Impagliazzo, 2004).
