We measured the phase-lag spectrum of the high-frequency quasi-periodic oscillations (QPO) in the black hole systems (at QPO frequencies) GRS 1915+105 (35 Hz and 67 Hz), GRO J1655-40 (300 Hz and 450 Hz), XTE J1550-564 (180 Hz and 280 Hz), and IGR J17091-3624 (67 Hz). The lag spectra of the 67-Hz QPO in, respectively, GRS 1915+105 and IGR J17091-3624, and the 450-Hz QPO in GRO J1655-40 are hard (hard photons lag the soft ones) and consistent with each other, with the hard lags increasing with energy. On the contrary, the lags of the 35-Hz QPO in GRS 1915+105 are soft, with the lags becoming softer as the energy increases; the lag spectrum of the 35-Hz QPO is inconsistent with that of the 67-Hz QPO. The lags of the 300-Hz QPO in GRO J1655-40, and the 180-Hz and the 280-Hz QPO in XTE J1550-564 are independent of energy, consistent with each other and with being zero or slightly positive (hard lags). For GRO J1655-40 the lag spectrum of the 300-Hz QPO differs significantly from that of the 450-Hz QPOs. The similarity of the lag spectra of the 180-Hz and 280-Hz QPO in XTE J1550-564 suggests that these two are the same QPO seen at a different frequency in different observations. The lag spectrum of the 67-Hz QPO in GRS 1915+105 is significantly different from that of the $2.7 times 10^{-4}$ Hz QPO in the narrow-line Seyfert 1 galaxy RE J1034+396, which disproves the suggestion that the two QPOs are the same physical phenomenon with their frequencies scaled only by the black-hole mass. The lag spectrum of the QPO in RE J1034+396 is similar to that of the 35-Hz QPO in GRS 1915+105, although identifying these two QPOs as being the same physical feature remains problematic. We compare our results with those for the lags of the kilohertz QPOs in neutron-star systems and the broadband noise component in active galactic nuclei, and discuss possible scenarios for producing the lags in these systems.
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