We use full-disk soft X-ray data from Yohkoh and Kitt Peak daily magnetograms to study the coronal irradiance and photospheric magnetic field remote from active regions between 1991 November and 1998 December. For every image of our data set we extract three areas 4° × 4° in size centered at N00° W00°, N50° W00°, and S50° W00° and compute X-ray irradiance and unsigned magnetic flux for each of these areas. Between 1991 (active Sun) and 1996 (quiet Sun) the X-ray irradiance at the heliographic center decreased by more than a factor of 7, while the magnetic flux decreased by only a factor of 2. A similar tendency is observed for our high-latitude samples. Apart from the cycle-related variations, all three areas of quiet Sun exhibit significant nonperiodic changes in X-ray irradiance. These variations occur on 9-12 month intervals and clearly correlate with changes in sunspot activity. Similar variations are present in the total X-ray irradiance averaged over the solar disk. By contrast, the magnetic fluxes from the photosphere beneath these same areas show no corresponding variations on this time scale. In our opinion, coronal heating models based on the reconnection of quiet-Sun magnetic elements (variously called chromospheric network, "magnetic carpet," or "salt and pepper" field) can at best account for a minimal contribution to heating the million-degree corona observed by the Yohkoh soft X-ray telescope. We conclude that the X-ray irradiance in the quiet Sun (at least in the Yohkoh temperature range, >2 MK) is primarily associated with the strong magnetic fields of active regions, not with weak photospheric fields. The association, however, is not direct. We interpret the enhanced X-ray irradiance above the quiet Sun not as a result of the coronal "canopy" formed by the active-region magnetic field above the quiet-Sun areas, but as the large-scale relaxation process in the corona triggered by the evolution of active regions. To further support this conclusion, we show examples of active and quiet hemispheres in 1996 with similar weak-field properties but greatly different global X-ray emission and a pronounced change in X-ray irradiance over the entire visible hemisphere that was associated with the emergence of a single small active region.