Most galaxies contain supermassive black holes (SMBHs) heavier than 106M⊙ [1], and the existence of black holes (BHs) with masses between a few and ∼ 80M⊙ has been established by observations of X-ray binaries [2] and by the measurements of gravitational waves (GWs) with frequencies ∼ 100 Hz emitted during their mergers [3–5]. Various other observations point to the existence of intermediate-mass black holes (IMBHs) with masses in the range 104 − 106M⊙, but their mass function and redshift distribution is known only very poorly [6]. This lack of information about IMBHs impedes our understanding of how SMBHs have been assembled [7]. The main possibilities for seeding SMBH assembly include collapsed Population-III stars [8], BHs formed in nuclear star clusters via repeated mergers [9–11] or direct collapses of protogalactic disks in which fragmentation is suppressed [12–18]. All of these mechanisms are capable of reproducing the properties of the observed SMBH population for suitable values of assembly parameters such as accretion rates, but can differ significantly in their predictions for the spectrum of IMBH mergers at different redshifts, see [19] for an overview. Signatures of these mechanisms may include either light seeds, with masses ∼ 102 − 103M⊙ or heavy seeds, with masses ∼ 104 − 105M⊙, at z ≲ 10. These are currently unconstrained by data, but can in principle be probed by future GW and other measurements [20–26]. The purpose of this paper is to investigate what progress can be made in distinguishing between the SMBH assembly scenarios with planned future GW experiments in light of recent data on nHz GWs from pulsar timing array (PTA) experiments, highlighting the potential capabilities of atom interferometers. NANOGrav [27] and other PTA experiments [28–30] have recently reported the observation of a stochastic background of GWs at frequencies in the nHz range, for which the most conservative astrophysical interpretation is that binary systems of SMBHs are emitting them with masses ∼ 109M⊙ [31–34]. Naive extrapolation of binary merger models to lower BH masses suggests that GWs from IMBH binaries may be observable at higher frequencies between 10−5 Hz and 1 Hz, for example by the LISA space-borne laser interferometer experiment or atom interferometer experiments [35]. In principle, there are two regimes where the formation channels of the SMBHs may be distinguished. Either at z ≳ 7 when the seeds are assembling and scaling relations are not that strong [36, 37], or by observing the low mass occupation fraction of dwarf galaxies and dark matter halos at more recent times [19, 38]. The extrapolation to higher frequencies of a model that can fit the NANOGrav background [34] predicts that the majority of detectable binaries will be at z < 7, so the focus of this study is to constrain the latter. Our work complements analogous studies that have been performed with electromagnetic observations of active galactic nuclei (AGNs) [39–42].