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To characterize the computational complexity of satisfiability problems for probabilistic and causal reasoning within Pearl’s Causal Hierarchy, van der Zander, Bläser, and Liśkiewicz [IJCAI 2023] introduce a new natural class, named succ-∃ℝ. This class can be viewed as a succinct variant of the well-studied class ∃ℝ based on the Existential Theory of the Reals (ETR). Analogously to ∃ℝ, succ-∃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-∃ℝ in terms of nondeterministic real Random-Access Machines (RAMs) and develop structural complexity theoretic results for real RAMs, including translation and hierarchy theorems. Notably, we demonstrate the separation of ∃ℝ and succ-∃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-∃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-∃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 ∃ℝ^Σ 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. Furthermore, we study the satisfiability problem for probabilistic reasoning, with the additional requirement of a small model, and prove that this problem is complete for ∃ℝ^Σ.