In order to develop a long-lifetime MCP-PMT under high rates of circumstance, we investigated the degradation of the quantum efficiency (QE) of PMTs with a multialkali photocathode. We found that not only positive ions, but also neutral residual gases would damage the photocathode resulting in an enhancement of the work function; their countermeasures were established in newly manufactured square-shaped MCP-PMTs with 4 or 4x4 multi-anodes. The performances of the PMTs were measured: QE was stable up to an integrated amount of anode output charge of 2-3 C/cm^2, while keeping other basic performances steady, such as the time resolution for single photons of ~40 ps, a photoelectron collection efficiency (CE) of 60%, a multiplication gain (G) of a few x 10^6, and dark counts of 20-300 Hz. The causes of QE degradation are discussed.
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