  To characterize the computational complexity of satisfiability problems for
probabilistic and causal reasoning within the Pearl's Causal Hierarchy,
arXiv:2305.09508 [cs.AI] introduce a new natural class, named succ-$\exists$R.
This class can be viewed as a succinct variant of the well-studied class
$\exists$R based on the Existential Theory of the Reals (ETR). Analogously to
$\exists$R, succ-$\exists$R is an intermediate class between NEXP and EXPSPACE,
the exponential versions of NP and PSPACE. The main contributions of this work
are threefold. Firstly, we characterize the class succ-$\exists$R in terms of
nondeterministic real RAM machines and develop structural complexity theoretic
results for real RAMs, including translation and hierarchy theorems. Notably,
we demonstrate the separation of $\exists$R and succ-$\exists$R. Secondly, we
examine the complexity of model checking and satisfiability of fragments of
existential second-order logic and probabilistic independence logic. We show
succ-$\exists$R- completeness of several of these problems, for which the
best-known complexity lower and upper bounds were previously NEXP-hardness and
EXPSPACE, respectively. Thirdly, while succ-$\exists$R is characterized in
terms of ordinary (non-succinct) ETR instances enriched by exponential sums and
a mechanism to index exponentially many variables, in this paper, we prove that
when only exponential sums are added, the corresponding class
$\exists$R^{\Sigma} is contained in PSPACE. We conjecture that this inclusion
is strict, as this class is equivalent to adding a VNP-oracle to a polynomial
time nondeterministic real RAM. Conversely, the addition of exponential
products to ETR, yields PSPACE. Additionally, we study the satisfiability
problem for probabilistic reasoning, with the additional requirement of a small
model and prove that this problem is complete for $\exists$R^{\Sigma}.
