  The problem $\textrm{PosSLP}$ involves determining whether an integer
computed by a given straight-line program is positive. This problem has
attracted considerable attention within the field of computational complexity
as it provides a complete characterization of the complexity associated with
numerical computation. However, non-trivial lower bounds for $\textrm{PosSLP}$
remain unknown. In this paper, we demonstrate that $\textrm{PosSLP} \in
\textrm{BPP}$ would imply that $\textrm{NP} \subseteq \textrm{BPP}$, under the
assumption of a conjecture concerning the complexity of the radical of a
polynomial proposed by Dutta, Saxena, and Sinhababu (STOC'2018). Our proof
builds upon the established $\textrm{NP}$-hardness of determining if a
univariate polynomial computed by an SLP has a real root, as demonstrated by
Perrucci and Sabia (JDA'2005).
  Therefore, our lower bound for $\textrm{PosSLP}$ represents a significant
advancement in understanding the complexity of this problem. It constitutes the
first non-trivial lower bound for $\textrm{PosSLP}$ , albeit conditionally.
Additionally, we show that counting the real roots of an integer univariate
polynomial, given as input by a straight-line program, is $\#\textrm{P}$-hard.
