<p>Bosonic quantum systems operate in an infinite-dimensional Hilbert space, unlike discrete-variable quantum systems. This distinct mathematical structure leads to fundamental differences in quantum information processing, such as exponentially greater complexity of state tomography¹ and factoring in constant space². Yet, it remains unclear whether this structural difference may translate to a practical computational advantage over finite-dimensional quantum computers. Here we take a step towards answering this question by showing that universal bosonic quantum computations can be simulated in polynomial space (and exponential time) on a classical computer, improving the previous best upper bound requiring exponential memory³. In complexity-theoretic terms, we improve the best upper bound on CVBQP with at most exponential energy from EXPSPACE to PSPACE. This result is achieved using a simulation strategy based on finite energy cutoffs and approximate coherent state decompositions. While we propose ways to potentially refine this bound, we also present arguments supporting the plausibility of an exponential computational advantage of bosonic quantum computers over their discrete-variable counterparts. Furthermore, we emphasize the role of circuit energy as a resource and discuss why it may act as the fundamental bottleneck in realizing this advantage in practice.</p>