Context. The vertical current sheet (VCS) trailing coronal mass ejections (CMEs) is the key place at which the flare energy release and the CME buildup take place through magnetic reconnection. The VCS is often studied from the edge-on perspective for the morphological similarity with the two-dimensional “standard” picture, but its three-dimensional structure can only be revealed when the flare arcade is observed side on. The structure and dynamics in the so-called supra-arcade region thus contain important clues to the physical processes in flares and CMEs. Aims. We focus on supra-arcade spikes (SASs), interpreted as the VCS viewed side on, to study their spatiotemporal structures. By comparing the number of spikes and the in situ derived magnetic twist in interplanetary CMEs (ICMEs), we intend to check on the inference from the standard picture that each spike represents an active reconnection site and that each episode of reconnection adds approximately one turn of twist to the CME flux rope. Methods. For this investigation we selected four events, in which the flare arcade has a significant north-south orientation and the associated CME is traversed by a near-Earth spacecraft. We studied the SASs using high-cadence high-resolution 131 Å images from the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory. Results. By identifying each individual spike during the decay phase of the selected eruptive flares, we found that the widths of spikes are log-normal distributed. However, the Fourier power spectra of the overall supra-arcade extreme ultraviolet emission, including bright spikes, dark downflows, and the diffuse background, are power-law distributed in terms of either spatial frequency k or temporal frequency ν, which reflects the fragmentation of the VCS. We demonstrate that coronal emission-line intensity observations dominated by Kolmogorov turbulence would exhibit a power spectrum of E(k) ∼ k−13/3 or E(ν) ∼ ν−7/2, which is consistent with our observations. By comparing the number of SASs and the turns of field lines as derived from the ICMEs, we found a consistent axial length of ∼3.5 AU for three events with a CME speed of ∼1000 km s−1 in the inner heliosphere; but we found a much longer axial length (∼8 AU) for the fourth event with an exceptionally fast CME speed of ∼1500 km s−1, suggesting that when the spacecraft traversed its leg this ICME was flattened and its “nose” was significantly past the Earth.